NTIA-REPORT-80-34

Three Phases of

TeletrafficCongestion in

Military Access Areas

M. Nesenbergs

R.F. Linfield

Prepared for

U.S. Army Communications Systems Agency

Ft. Monmouth. New Jersey 07703

u.s. DEPARTMENT OF COMMERCE

Philip M. Klutznick, Secretary,

Henry Geller, Assistant Secretary

for Communications and 'Information

February 1980
DISCLAIMER

The views, opinions, and/or findings contained in this

report are those of the authors and should not be construed

as an official Department of the Army position, policy, or

decision unless designated by other official documentation.

i i
PREFACE

The study reported here was performed for the U. S. Army's

Communications Systems Agency (CSA), Fort Monmouth, NJ, by the

Institute for Telecommunication Sciences (ITS), Boulder, CO.

It is part of a continuing program in support of that agency's

Access Area Digital Switching System (AADSS) program on project

orders 50l-RD, 804-RD, 809-RD, and 906-RD.

Previous related study reports are as follows:

"Parametric Cost Alternatives for Local Digital Distribution

Systems,1I M. Nesenbergs and R. F. Linfield, OT Report 76-95,

March 1976 ..

"Preliminary Evaluation of Hub Alternatives for Access Area

Digital Switching," J. C. Blair, Special unpublished ITS

report to CSA, October 1977.

"Access Area Switching and Signaling: Concepts, Issues, and

Alternatives," R. F. Linfield and M. Nesenbergs, NTIA

Report 78-2, August 1978.

"Control Signaling in a Military Swi tchinq Environment,"

R~ Fw. Linfield, NTIA Report 79-13, January 1979.

"Switch Element Capacities in Access Area Digital Switching

Systems," R.. F" Linfield and M. Nesenbergs, NTIA Report

79-26, September 1979.

Administration and technical monitoring of this current

study contract was performed by Mr. T. Michelli of CSA. Technical

and management supervision of the program at ITS was provided by

Dr. P. M. McManamon,

iii

LIST OF FIGURES

LIST OF TABLES

ABSTRACT

TABLE OF CONTENTS

v

Page

vi

vii

1
LIST OF FIGURES

Page

Phase 1 - blocking at a concentrator. 8

Basic deployment concept of a military access area. 2

Three phases of blocking. 7

Approximate Phase 1 blocking probabilities for M=12 sources. 16

Approximate Phase 1 blocking probabilities for M=24 sources. 17

Approximate Phase blocking probabilities for M=48 sources. 18

Approximate Phase blocking probabilities for M=96 sources. 19

Exact Phase blocking probabilities for an infinite number

of sources. 20

35

33

23

24

26

The effect of the number of users, M, on 1% out/in call

blocking.

Phase 2 - blocking at a PABX.

Application examples of the Phase 2 model.

The general effect of the Engset distribution on the

blocking probability for Phase 2.

Figure 13. Phase 2 blocking probabilities for loop-to-line ratio

m/n=1/2.

Figure 1•

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11 .,

Figure 12.

Figure 14.

Figure 15.

Figure 16.

Figure 17.

Figure 18..

Phase 3 - blocking at a switch. 36

The three blocking probabilities of Phase 3. 44

Phase 3 blocking for a symmetric switch (L=T) with a

symmetric load (B=C). 46

Hierarchical network configurations for interregional access. 53

Implementation concept for digital concentration, multiplexing,

switching and transmission in the access area. 57

Figure 19. Local and remoted components of a switching hub. 58

Figure 20.. Illustrative example of an access area communications network. 60

Figure 21. Sizing of line and trunk groups in the three phase format. 63

vi
Table 1.

Table 2.

Table 3.

Table 4.

Table 5.

Tabl e 6.

Table 7.

Table 8.

Table 9.

Table 10.

Table 11.

Table 12.

Table 13.

LIST OF TABLES

Various Route Profiles

Column Vector C for Different (i,j) States

Column Vectors C (v=1,2) for Different (i,j) States

-\)

Column Vector I for the (i,j) States of Blocking Phase 2

Notation for the Three Kinds of Service Requests from Two

Classes of Server Channels

The Seven State Regions and Their Blocking Roles

The Column Vectors for the Seven State Regions k=1,2, ... ,7

Blocking Probability Structures for the Forty Access Area

Circuit Paths

Estimates of Military Communications Terminals, Local to

Worldwide

Telephone Terminal Density Profile

Main Line and Bridged Telephone Distribution Profile

Assumed Loads per Line (in Erlangs) for Selected Terminalto-Terminal Group Substreams

Blocking Probabilities for All Forty Paths for the Assumed

Access Area

vii

Page

5

11

13

28

39

40

42

48

52

55

56

61

65

THREE PHASES OF TELETRAFFIC CONGESTION IN MILITARY ACCESS AREAS

Martin Nesenbergs and Robert F. Linfield*

In military base access areas, akin to many commercial

installations, the telecommunications traffic passes through

several concentration and switching stages. Loads to server

facilities are formed through mergers and branch outs of

offered traffic substreams. Blocking of calls is known to

occur in many ways throughout the existing access area networks.

In this report, an effort is made to represent the access

area grades of service (i.e., the probabilities of blocking

for different substreams) in more realistic ways than before.

The message flow process is structured into three representative contention phases. The three phases are realistic and

occur often in military networks. All three phases apparently

possess queueing models and analytical properties distinct from

the conventional Engset, Erlang, and other classical models.

Their blocking probabilities also differ significantly.

One of the three models appears tractable only through

bounds and asymptotically tight approximations. The other two

models are shown to permit formal solutions. Given an access

area network, the three blocking probabilities may be applied

individually or in a variety of combinations. The paper demonstrates several applications to access area telephony.

1. INTRODUCTION

The military access area telecommunications traffic possesses numerous characteristics. Perhaps of most visibility, are the impressive types of communication

services provided to various terminal mixes of end-users. Of comparable importance,

if not always to the system designers and large network managers, but certainly to

individual users, is the quality of service that the individual user perceives.

This paper is concerned with one dimension of service quality, the so-called

"grade of service". In the present context, grade of service is interpreted as a

probability level of being blocked by the system. Thus, busy called parties and

malfunctioning message receivers are ignored. This study further assumes that at

least one end user of the communication transaction resides in a military access

area.

The access area concept is illustrated in Figure 1 (Krevsky, et al., 1972;

Levine, 1976). This figure stresses three interacting parts: the military access J

area (which is of primary interest to this report), the Defense Communications

*The authors are with U. S. Department of Commerce, National Telecommunications and

Information Administration, Institute for Telecommunication Sciences, Boulder,

Colorado 80303.
MUX/CONC

END

OFFICE

REMOTE

HUB

END

OFFICE

REMOTE HUB

Des Backbone

Terminals Access Area

Figure 1. Basic deployment concept ofa military access area.

2
System (DCS) backbone network, and the domestic common carriers (Linfield and

Nesenbergs, 1978; Linfield, 1979).

The access area proper may be comprised of one or two, but not many more,

military bases. The access area bases share parts of a common telecommunications

network (Nesenbergs and Linfield, 1976). In addition to transmission facilities,

the access area may contain central switching hubs (or switches, or hubs, for

short), lesser switches (such as PABX·s or PBX·s), plus various concentration

elements. Both multiplexers (MUX) and intelligent or statistical concentrators

(CONC) are remotable devices found in cost effective designs of access area

networks.

Externally, the access area interfaces with the previously mentioned DCS

backbone and with the common carriers. The access area telecommunications traffic

is either generated by the user terminals in that area or i,s destined for the

terminals in the area. Seldom, if ever, does a military access area network play

a tandem role for through traffic.

The access area traffic may be partly local, i.e., going to the same or to a

geographically adjacent site, or it may be destined for some distant military

region. In the latter case, the traffic is switched outward from or inward to

the access area, either on the DCS backbone or via the common carriers. At the

other end, the terminating address could represent a military terminal or a

civilian (government, commercial, household, or institutional) installation.

When a circuit path is established, it can take one of several routes through

the cooperating networks. The route may pass through the DCS, but not more than

once. It may pass through the toll or local facilities of a common carrier, but

also with high probability, not more than once. And, to be complete, at least

once but certainly not more than twice, the route must traverse local access

areas.

Given that individual military communications must take one of such paths,

several traffic engineering questions arise immediately. Does the existing traffic

engineering theory provide needed tools for access area system design of the 1980·s?

The classic work of Erlang (Brockmeyer, et al., 1948), Molina (1927), Fry (1928),

and others provides a basic background. However, it hardly extends into the

specifics of modern switched networks. More realism is needed. That realism is

partly found in the neo-classic models, of the fifties and sixties (Jacobeaus, 1950;

Lee, 1955; RBnnblom, 1959; Lotze, 1963; Wolman, 1965; Lee and Brzozowski, 1966).

Unfortunately, the complexity encountered by these writers is quite formidable.

Their results appear difficult to interpret and to apply. It is therefore not

3
surprlslng that recent workers have generated a r~surgence of useful switched

network characterization (Kuczura and Bajaj, 1977; Lin, et al., 1978; Nesenbergs,

1979; Manfield and Downs, 1979; Haugen, 1979). The present study follows in their

wake by seeking reasonable accuracy, simplicity, and applicability to the access

area.

Succinctly put, this report addresses the teletraffic blocking problem for a

circuit switched access area. A simple and apparently effective sequential circuit

flow description is developed. Starting with the next section, a set of three

congestion or blocking phases are introduced. The intent is to represent the

acfess area blocking events as uniquely identifiable, separate members of these

blocking phases.

Thereafter, in later sections~probabilityof blocking formulas and curves

are developed for the three phases (see Sections 3, 4, and 5). Section 6 sunmarizes

these analytical and numerical tools by applying them to representative access

area scenarios.

2. THREE PHASES OF BLOCKING IN ACCESS AREA

The previous section has alluded to the variety of routes that access area

traffic may take. If so, then network blocking may occur in a number of ways and

in a number of places.

Let us be more specific. Assume that the DCSbackbone and common carrier nets

have sufficient capacity to be essentially nonblocking. Blocking - if it occurs -

must then occur within an access area. But there, as suggested in Figure 1, the

route may encounter three kinds of nodal elements: the concentrator (CONC), the

remote hub (or the small customized switch, or PABX) , and the larger central hub

or the dial central office (DCO). The latter is also variously referred to as the

switch, the end office (EO), the local telephone exchange, or the Class 5 switching

center.

In the originating access area, the route may pass through various combinations

of CONC, PABX, and switches. The same is true at the destination access area, if

a separate such area is involved. All total, some forty different route profiles

may occur. They are tabulated in Table 1. Each row in this table represents an

access area communications path. For instance, path #1 is one routed through a

concentrator, followed by a PABX, plus the switch, all in the initial access area.

In the final access area, as shown, it again passes through another switch, another

PABX, and another concentrator.

4
Table 1. Various Route Profiles

Initial Additional

Path

I # CONC PABX Switch Switch PABX CONC !

!

i 1 X X X X X X

I 2 X X X X X I

3 X X X X X I

I

4 X X X X

5 X X X X X

6 X X X X

7 X X X X

8 X X X

I

9 X X X X X

10 X X X X

11 X X X X

12 X X X

13 X X X· X

14 X X X , ,... X v v I:)

I

A 1\

16 X X

17 X X X X X

18 X X X X

19 X X X X

20 X X X

21 X X X X

22 X X X

23 X X X

24 X X

25 X X X X

26 X X X

27 X X X

28 X X

29 X X X

30 X X

31 X X

32 X

33 X X X X

34 X X X

35 X I X X

36 X X

37 X X X

38 X I X

39 X X

40 X I ----._-_._-- --

5
The paths in Table 1 ignore everything that happens in the DCS backbone or in

the commercial common carriers. For simplicity sake, they also ignore several

local elements, such as operator consoles (often blocking), multiplexers (nonblocking), and others of lesser significance.

Nevertheless, the number of route profiles is considerable. And for any offered

load distribution for different services, each of the forty paths is apt to have

different grade of service characteristics. The task is considerably simplified

if one can assign a unique probability of blocking feature to the three elements:

the CONC, the PABX and the switch. In what follows, such a three-step approach is

introduced.

Consider Figure 2. It defines the three phases of circuit blocking. The

first phase Of blocking occurs at a concentrator. Analytically, this will turn

out to be the most unwieldy of the blocking phases. The complete problem setting

of blocking Phase 1 and its approximate solution are presented in Section 3.

The second phase of blocking in Figure 2 is associated with a PABX. Looking

from the user terminal side, the small PABX switch could be the first switching

entity encountered by a station line. Lines can pass through concentrators or

they canby-pass the concentrators and be directly patched to the PABX. The

probability of blocking for Phase 2 is treated in Section 4.

The third and last phase of blocking is shown (Figure 2) to take place at a

central switch. User terminals can be connected directly to. the switch. Or they

can be routed either through the concentrator, or the PABX, or both. The switch

also has a long distance or toll side. For example, the right hand side of the

switch in Figure 2 can be viewed as the collection of trunks to other DCS or

common carrier switches. Section 5 discusses the Phase 3 blocking that occurs at

such a switch.

3. BLOCKING PHASE 1

3.1. Problem Definition

Assume the m-server facility of Figure 3, where two types of calls contend for

the services of a concentrator. In a nutshell, the two types are distinguished

by the number of servers (e.g., one or two) that they require per call.

The users or sources are the Mterminals. Typically, M>m. To simplify

matters t assume exponential distributions for both the interarrival and service

(holding) times. Let blocked calls be lost or cleared without any aftereffect.

Let the average service times be unity for both call categories, denoted as (ex)

6
........J

, , y

Blocking

Phase 1

, , ...

Blocking

Phase 2

Figure 2. Three phases of blocking.

, I "..

Blocking

Phase :3
PABX SWITCH

m

M

----... CONCEN- I I

I TRATOR

, v

Blocking

Phase 1

Figure 3. Phase 1 - blocking at a concentrator.
(1 )

and (8) in Figure 3. In an informal way, one may refer to the (a) calls as

"distant" calls, and to (8) calls as IIlocalli. However, one should note that (a)

represents a set of paths with common properties. In Table 1, (a) is identified

with any path that passes through a concentrator, except for #36. Path (8) depicts

only #36.

Assume that the average arrival rates are different for the distant and local

calls. Let lIall be the arrival rate per,idle terminal for distant or (a) calls,

and let "b" be the corresponding rate for local or (8) calls. Finally, assume

that the concentrator and the PABX have full availability and that the PABX is

effectively nonblocking, at least as far as the present Phase 1 blocking is concerned.

The problem here is to derive the blocking probabilities for the two types of

calls. The given parameters are M, m, a and b. This is a queueing system or a

birth-death problem of the M/M/m/O/M kind (Kleinrock, 1975). The next-to-last 0

refers to the lack of buffer storage in the assumed loss system, and the final M

represents the finite user population. Systems of this type are known to have

the desired steady state or equilibrium properties (Kleinrock, 1975; Saaty, 1961;

Riordan, 1962). All state probabilities, including blocking probabilities, do

exist. The usual derivation of state probabilities employs the flow rate conservation or statistical equilibrium difference equations. The same approach will be

pursued here.

Let the integer pair (i,j) stand for the joint event or state, that i distant

(a) and j local (8) calls are in progress. Let the equilibrium probability of

state (iltj) be p(i,j). Then

') p(i,j) = 1.

au i ,j

To get rid of trivial terms, require that p(i,j) vanishes for all i<O, all j<O, as

well as for all i+2j>m.

Blocking probabilittes are of interest here. Whenever i+2j=m, outgoing and

incoming calls of type (a) are blocked. Likewise, the local return or (8) calls

are blocked if and only if' i+2j>m-l. One defines the respective blocking probabilities aS t

I p(i,j), i+2j=m

p = r

I p(i,j), i+2j~m-l

9

(2)
where subscript 0 depicts the "out/inll nature of the (a) category, and r stands

for the "return ll or (S) calls.

For each (i,j), define the following column vector of state probabilities:

p(i-l,j)

p(;,j-l)

p = P(i, j ) (3)

p(i+l,j)

p{i,j+l)

Vectors play the role of unknowns. They must be solved using flow or transitionrate conservation equations. There are many such equations, applicable to different

(i,j), and all of the form

T

C - P = O. (4)

Here, fT is the transpose of the column vector C. Vector f. consists of the appropriate flow rate or birth-death coefficients needed to conserve the outward and

inward flows at the state (i,j). The dot (-)in (4) stands for the ordinary inner

product of two five-dimensional vectors.

The coefficient vector f is given in Table 2. The columns in the table are

the column vectors. They depend on the system state (i,j) as shown by the three

different columns. For non-vanishing p(i,j), one considers only (i,j) that satisfy

m;n{i,j,m-i-2j}~0, as noted earlier. This obviates the need for special i=O and

j=O columns, where the first and the second rows vanish, respectively and by definition.

Table 2 is the initial coefficient listing of this report. Others, such as

Tables 3,4 and 7, will follow. It seems helpful, at least initially, to explain

the structure of a typical vector f... Assume i+2j2..fl1-2, which corresponds to the

left column of Table 2. The easiest place to begin maybe the third row. It

depicts~ see (3), the state (;,j) itself.

Given that the system is in the state (i,j), outward flows may be caused by

several mechanisms. First, any of the i active (a) calls may hang up. Due to

unity holding time, this happens with average rate i. Second, the j (6) calls

cease with average rate j. Next, there are the two types of call arrivals. Since

for the (i,j) state M-i-2j users are idle, they contribute (M-i-2j)a (a) service

requests per unit time. Likewise, on the average, they generate (M-i-2j)b calls

10
Table 2. Column Vector C for Different (i,j) States

i > 0 , j > 0 -

i+2j -

< m-2 i+2j = m-l i+2j = m

- (M- i - 2j+1)a -(M-m+2)a -(M-m+l)a

-(M-i-2j+2)b - (~1-m+3) b -(M-m+2)b

i+j+(M-i-2j)(a+b) i+j+(M-m+l )a i+j

-(i+1 ) -(i+l) 0

- (j+1) 0 a

11
of the (8) type. The total average outflow from the (i,j) state is their sum,

namely i+j+(M-i-2j)(a+b). It is the coefficient entry in the third row, first

column, of Table 2.

The fi~st row shows the flow into state (i,j) from state (i-l,j), see (3).

The i increment is generated by (a) traffic from M-i-2j+l idle sources. Their

total \average contribution is thus (M-i-2j+l)a. In Table 2 this entry carries a

negative sign, because inflow counteracts the outflow.

The second row of Table 2 represents the (i,j) inflow from the state (i,j-l).

It is a (S) flow from M-i-2j+2 idle users. The resultant coefficient in the second

row is thus -(M-i-2j+2)b.

The fourth and fifth rows are similar. They give the inflows to state (i,j)

from states (i+l,j) and (i,j+l), respectively. In both cases, the assumed unity

average service time dictates that the coefficients are -(i+l) and -(j+l).

The net sum of the positive and negative products must vanish, see (4), to

ensure flow equilibrium. That is the steady state solution p(i,j) for all i and j

(Kleinrock, 1975, Chap. 3).

The solution to the equation set (3), (4), plus Table 2, appears unknown and

difficult. To simplify, assume with no great loss of generality that both M and m

are even numbers. Then one has (~1)2 unknowns, p(i,j), as well as the same

number of equations (4). The set of equations turns out to be linearly dependent.

Fortunately, that is a minor problem. Any equation may be replaced by the normalization condition (1).

A bigger problem is the number of equations. Consider a typical Tl or T2

span between the concentrator and the PABX. The Tl accommodates 24 PCM channels,

the T2 up to 96. In the first case, the number of equations is 169, in the second

case it is 2401. Inversion of matrices of such high dimensionality is not easy,

even with the latest computer resources. Moreover, a single matrix inversion provides numerical ans~ers only for the selected values of M, m, a and b. To depict

acomprehensive'grade of service picture, many tedious computer runs may be needed.

To proceed, a different approach is tried here. An approximate model is proposed

for Phase 1. The model provides the desired tight asymptotic fits plus bounds on

blocking probability. The model is tractable and is given next.

3.2. The Approximating Model

Instead of the previous exact coefficient vector f, consider two approximate

vectors, C , where v=1,2. Let thee vectors be defined by the state dependent -v -v

columns of Table 3. As in Table 2, the first row vanishes when i=O, the second

vanishes when j=O.

12
Table 3. Column Vectors C (v=1,2) for Different (i,j) States

-v

i > 0 , j > 0

i +2j < m-2 i +2j = m-l i +2j = m -

- (M-vi-vj +v)a -(M-vi-vj+v)a - (M -vi-vj +v)a

-(M-vi-vj+v)b - (~1-vi -vj+v) b - (M-v;-vj+v) b

i+j+(M-vi-vj)(a+b) i+j+(~1-v;-vj)a i +j

-(i+l ) -(i+l ) 0

-(j+l) 0 0 /

13
The solutions generated by the approximate flow coefficient model, ~, will

be denoted as p (i,j). In general, state probabilities p (i,j) must differ from

v v

the exact p(i,j). However, there are at least two advantages for looking at the

p (i,j)'s. First, one obtains closed form expressions. And second, in asymptotic \J

and limit cases of interest the expressions agree with the true p(i,j).

The solution to equations (1), (3), (4), and Table 3 is given by

(5)

where v=1,2, and m;n{i,j,m-i-2j}~O. The proof is omitted here for reasons of

brevity. It can be carried out by substituting (5), for all three cases of Table

3, into (1), (3) and (4). The procedure is long" but straightforward.

The asymptotic behavior of the approximating solutions (5) turns out to be

related to the Engset distribution (Riordan, 1962, Chap. 5):

(6)

where k=0,1,2, ... ,m.

Consider first the case when the distant (a) traffic becomes negligible.

Then the only nonvanishing terms occur at i=O, and

p(O,j) ---...E. C~~;~,2b) , a-+O J

p,\(O,j) - .......E.(!!!)1,vb). v a-+O J v v

(7)

Thus, v=2 offers the correct asymptotic fit for a-+O. Next, assume that the local

(s) traffic vanishes. Then j=O is of interest and

p(i ,0) - ......E. (m;M,a), b-+O 1

-~.E (m.M ) i -,-,va · b-+O v v

(8)

Now, contrary to the previous case, v=l gives the correct fit to the true p(i,j).

14
Finally~ one must not exclude the practically significant limit of infinitely

many sources. One lets M~, while insisting that Ma+A and Mb+B. Then, as is

easily shown (R5nnb1om, 1959), all solutions tend to the same limit:

(9)

AiBj

-·-1·-1

1 •J ·

I ASBt ·

all s,t s!t! I

p(i , j ) I---.-----=----

pv(i,j)

This last, M+oo, limit distribution is not Engset either. However, it does provide

a common correct limit for both v=l and v=2.

From the asymptotic properties, (7) to (9), one feels justified to suggest a

II rea1i st ic" approximation,

p(i,j) = a~bPl(i,j) + a~bP2(i,j). (10)

Clearly, p(i,j)+p(i,j) for all a+O, b+O, and M~ cases.

The behavior of this approximation, plus possible bounds, will be explored in

the next section on numerical results.

3.3. Numerical Results

Based on equations (2) to (10), the out/in and return call probabilities of

blocking are approximately,

L p(i,j), i+2j=m

I i+2j~m-l

(11 )

In this approximation, the quantities p(i,j) are given in terms of pv(i,j)

110). !he latter pv(i,j) are defined in (5). With the aid of'these definitions,

Po and Pr have been computed for selected values of parameters M, m, a and b. The

results are presented in Figures 4 to 8.

All of these figures consist of four parts. The four parts show constant '"

blocking probabilities, such as P =0.002 or P =0.010 for distant out/in calls, and a 0

Pr=0. 010 or P

r

=Oe050 for local return calls. The abscissa in all cases is m, the

number of server channels. Notice that the m scale may be different for the

different figures, but it is consistently the same for the four parts of each

figure. Similarly, a common ordinate M(a+b) applies for every figure.

15
3 4 5 6

M=12

%=0.010

M=12

--'

I}=0.050

6 2

m

3 4 5

M=12 ,....,

Po=0.002

M=12

2.0 Pr=O.OIO

o 2

1.0

1.5

1.0

1.5

0.5

I 0.5

c

a

~O

:0 2.5

+

o...........

2.0

Figure 4. Approximate Phase 'I blocking probabilities for

f1= 12 SQU1"'ces.

16
6 8 10 12

M=24

~=0.050

a~b =1.0~ . =0.9~"

=0.5~

=0.-......." ... "

M=24

~=O.OIO

a~b~6~9~ =0.5~

=O~

10 12 2 4

m

6 8

M=24 ,...,.,

Po=0.002

~-IO +b -. ------------ . a =0.9_____.

=0.5~

=O~ <,

M=24

~=0.010

2

6

4

I0r---~--r----""--~-----' r--....,.---r-----...--~____.

8

6

r-----1 4

(/)

CJ)

§ 2

0I.....=::~~C---L_L-~ ~....-..----~-----.lI.-..-~--"-----'

:n 10

+

..g 8

Figure 5. Approximate Phase 1 blocking probabilities for

r1:.::24 sources.

17
M=48

~=0.050

M=48

~=O.oIO

a~b =1.0

=0.9

=0.5' ,

=0'

a

+b=LO~

a =0.9~"

=0.5",

-0 ' - " " \

M=48 "-

fb=0.002

15

10

25---..-------r--....------.r--------.

M=48

20 ~=O'OIO

O2 5 10 15 20 25 2 5 10 15 20 25

m

,...---,

(J)

01

§ 5

~O~==-----l..------L_---L.------J

:0- 25--------.----...---

+

o

--- 20

15

10

5

Figure 6. Approximate Phase 1 blocking probabilities for

t1:::48 sources.

18
M=96

Po=0.010

=1.0

=0.9

=0.5~ =O~

M=96

~=0.050

a

a+b =1.0___.. =0.9--

=05~

=~

M=96

Po=0.002

a~b =1.0

=0.9

=0.5~ =O~ '

O2 10 20 30 40 50 2 10 20 30 40 50

m

80 r---~-.----r--~-­

M=96

60 F}=o.OIO

40 a b: 1.0 . -09 '" =0.5~ o 20 =0",

0.....~=---L-----JL-----l.._---I

n 80

+

..s 60

1

40

20

Figure 7. Approximate Phase 1 blocking probabilities for

t1=96 sources.

19
IM=ro I Po=O.OIO

A

A+S=1.0--"'"

=0.9

=0.5~

=0-- <,

20

10

O2 10 a). 30 40 50 2 10 20 30 40 50

m

40 r----r-----r-----,--~____.

M=ro

30 Ff=o.olo

20 A~s=I.o~ =0.9

=0.5--- , =0-

r---'1

10

c:

o

W 0~--,-------a.-.._..L...-----'------.J

L-J 40

rn

; 30

Figure 8. Exact Phase 1 blocking probabilities for an infinite

number of sources.

20
The ordinate, M(a+b), deserves a comment. It is called the "initial" offered

load, as it is strictly observed only when i=j=O. To each initial load, there

corresponds the effective or measurable offered load. The effective offered load

may be defined as

L = (a+b)[M - I (i+2j)p(i,j)].

a11 i,j

(12)

The effective load is always less than the initial load, but at the same time it

is slightly higher than the actual carried load (Bear, 1976; Siemens, 1974).

If one prefers to present the blocking probabilities as functions of the

effective load, one has to carry out the corresponding additional calculation

(12). That calculation is skipped here.

Every quadrant part of each figure, Figures 4 to 8, contains four curves.

These curves are selected for constant values of parameter a/(a+b). The latter is

the long term average fraction of distant calls made through the service facility.

One should comment on the adjective "approximate ll in the captions of Figures 4 to

7. It is valid only for O<a/(a+b)<l. When either a-+O orb-+O, the curves become

exact, see equations (7), (8), and (10). The latter cases are also useful as

upper and lower performance bounds.

As the fraction of distant calls is increased, the same number of m servers

can handle more traffic in the sense of the initial load M(a+b). Such a statement, -

of course~ presumes a fixed grade of service, for instance P

r=O.010.

As the number of users, M, increases, a clear cut limit behavior is achieved.

In the limit, M~, ~he two types of initial load tend to their respective limits,

Ma-+A and Mb~B. The initial and effective loads become indistinguishable. All

the approximations converge to the correct values, and Figure 8 applies. Note

that the ordinate is the total load A+B.

In the computation of these blocking probability curves, certain cases can be

verified with the aid of published tables. Thui, the a/(a+b)=Q and a/(a+b)=l cases

correspond to two distinct Engset distributions; see earlier equations (7) and

(8). In addition to the Engset traffic tables (Siemens, 1974), one can use binomial

tables, such as the NBS binomial tables (1949). When~, the ordinary Engset

distributions tend to Erlang B. There are several Erlang B charts and tables

available (Siemens, 1974; AT&T, 1961), as well as the related rather extensive

Poisson tables (General Electric, 1962).

Figures 4 to 8 can be used in various ways. Given user groups and their

traffic statistics, the concentration networks can be economically designed. Or

given m, the number of concentrated channels, acceptable size terminal groups can

21
be configured. Likewise, the grades of service for out/in and return calls (i.e.,

blocking probabilities P and P) can be variously engineered. More on this

a r

subject will be said in Section 6. To emphasize the role of specific parameters,

the curves can be redrawn as desired. An example is shown in Figure 9.

Figure 9 assumes constants Po=O.OlO and a/(a+b)=O.9. The 1% blocking loss for

distant outgoing and incoming calls may be a typical service objective. Likewise,

the premise that 10% of all calls are turned around locally and 90% are not, may

be true in many military installations. Given these ground rules, plus a fixed

initial offered load M(a+b), one sees how many more servers are needed for larger

M. Most servers, m, are required when M=oo. This is not surprising, as finite

user populations tend to offer less load when a significant fraction of the terminals are occupied.

Other graphs, similar to Figure 9, can b~ deduced from the basic curves

(Figures 4 to 8) as needed. One must, however, be careful with the scales of the

variables involved. For instance, both M(a+b) and m for M=12 are limited to

small regions in Figure 9.

4. BLOCKING PHASE 2

4.1. Problem Definition

Assume the N-server facility of Figure 10, which is depicted to be a PABX.

Two groups of users~ of arbitrary sizes m and n and with different traffic, contend

for the N server channels shown.

Consider the two user classes. In accordance with the notation-of Figure 3,

the upper class offers traffic of type (a). The m users here are concentrator or

multiplex (MUX) "loops.1I The lower class of n users supplies traffic of the type

('1). The latter may be loosely called Illines" although there is nothtnq to forbid

these from being another class of concentrated loops. In practice, m~n. We

assume that min{m,n}~N. According to the tabulation of path profiles (see Table 1),

both classes (a) and ('1) may have many members. Thus, in Phase 2, class (a)

contains paths numbered 1,2,3, .... Class (y) consists of 9,10,11, ....

Assume exponential distributions for both the interarrival and service

(holding) times. Let the average arrival rate be ~l on every loop and let l/Ul be

its average service time. Likewise, assume that ~2 and ~2 are the average arrival

and service rates for the lines, respectively.

The reason for choosing different Al and ~2 is clear. Concentrated loops are

apt to carry more traffic intensity than individual station lines. The reason for

22
M=OO

(Exact)

~= 0.010

a

a+f)=0.9

30 40 50 60 70 80 90 100

m

10j M=24

M=I

I ..olll

00 10 20

70iii I I I I ii':» ,

60

(i) 50 C])

c

0

W 40

L..-....I

N

w :c 30

+

0

'---'

20

Figure 9. The effect of the number of users, M, on 1% out/in call blocking.
m N

I'~

-~

I CO NCEN- ----------1

~I TRATOR I I

n

, ,

Blocking

Phase 2

SWITCH

Figure 10. Phase 2 - blocking at a PABX.
permitting ~lr~2 has to do with potential generalization. One can envision many

uses and extensions of this problem in military switching network applications.

Two possibilities are shown in Figure 11. Part (A) of the figure depicts a common

pool of N service units, perhaps control and signaling units. Two types of links,

identified as lines and trunks, seek service from the common pool. The statistics

of the two service processes are likely to differ.

Part (B) of Figure 11 illustrates a version of the recently proposed voice

and data integration. Both actual and virtual circuit-switched voice may be concentrated on N common channels with data packets. Military applications have been

discussed by many (Vena and Coviello, 1975; Fischer and Harris, 1976; Weinstein,

et al.~ 1978; Schutzer, 1979; Fischer, 1979). There are also other nonmilitary

applications and issues (Sherman, 1971; Arthurs and Stuck, 1979; Tsuda, et al.,

1979; Schneider, 1979). Data packets are much shorter than a phone conversation

or any of its vocal segments. In practice, this model may have to be both further

generalized, or restricted, by incorporating military requirements, such as preemptive interrupts.

The generalization of the model for Phase 2 could also include alternate

options for blocked calls. That is, given that a service request finds all N

servers occupied, the request could be either lost (dropped), or cleared (rerouted),

or entered into a queue to wait its turn for available servers. The latter assuredly wbuld be the case for the data packets of Figure 11 (B). Access area

PABX's, for reasons of survivability, occassionally provide direct plus alternate

routes, at least between the busiest hubs.

The general topic is a mix of loss and delay system features. It leads to

blocking probabilities that are hybrids of Erlang Band C formulas, or variants

thereof (Nesenbergs, 1979). For the time being, however, assume that all blocked

Phase 2 calls are simply lost at the PABX. Finally, assume full availability for

all m loops and all n lines to the N service channels.

As done in Section 3, one next seeks the probability of blocking. It will be

seen shortly that the situation is rather fortuitous. A closed form solution will

follow.

Let (i,j) represent the event that i loops and j lines are receiving service.

Then O~i.::..m, O~j~n, and O<i+j~N must hold. The steady state probability of event

(i,j) is again denoted as p(i,j). These probabilities satisfy the same normalization condition of equation (1). Note, that there is a single, common blocking

event here. It applies for both (a) and (y) traffic and occurs when i+j=N. Thus,

25
(A)

Lines

(B )

FACILITY WITH N

CONTROL AND/OR • SIGNALING UNITS • •

• •

•

OJ OJ • • • CD

Trunks

Voice, ---------....t

Circuits _---""'II

Data

Packets ................. /

INTEGRATED

CIRCUIT AND

PACKET

CONCENTR ATOR

N

Common

-....--- Channels

Figure ll~ Application examples of the Phase 2 model.

2G
(13)

the probability of blocking, P for Phase 2 (subscript x to emphasize PABX) is x

given by

L p(i,j). i+j=N

To deduce the new state probabilities, p(i,j), one uses the same P vector

definition of equation (3), as well as the conservation of flow principle represented in (4). However, the coefficient vector f takes a different form now. The

new state dependent column vector f is given in Table 4, including its specific

i=O and j=O realizations.

4.2. The Solution

The solution p(i,j) to the equations (1), (3), (4), and Table 4, follows from

the separation of variables or product method (Cooper, 1972). Assume the proportionality,

(14)

for all nonnegative i and j, such that O~i+j~N. Upon substitution, this leads

to separate equations for the unknowns Pl(i) and P2(j).

Without going into the details, one obtains seven equations for the seven

columns of Table 4.

For i=O and j=O:

[mA,p,(O) - ~,p,(')J/p,(O)

+ [nA2P2(O) - ~2P2(')J/P2(O) = O.

For i=O and O<j<N:

[mA,p,(O) - ~,p,(')J/p,(O)

- [(n-j+l)A2P2(j-l) - j~2P2(j)JiP2(j)

+ [(n-j)A2P2(j) - (j+')~2P2(j+')J/P2(j) = O.

For i=O and j=N:

27
N

co

Table 4. Column Vector C for the (i,j) States of Blocking Phase 2

i

I i=O O<i<N i=N !

I

I j=O O<j<N j=N j=O O<j<N-i j=N-i j=O

0 0 0 -(m-i+l)Al -(m-i+l)Al -(m-i+l)Al - (m- N+1)Al

I

I 0 -(n-j+l)A2 -(n-N+l)A2 0 -(n-j+l )A2 -(n-j+l)A2 0

I mAl i]Jl +(m-i )A1 i]Jl+(m-i)Al I mAl i ]J1 N]Jl

I +nA2 +j 1J2+(n-j )A2 N]J2 +nA2 +j]J2+(n-j)A2 +j 1J2

-111 -]Jl 0 -(i+l )]Jl -(i+l)]Jl 0 0

-]J2 - (j +1)1-12 0 - 1J2 -(j+l) 1J2 0 0

r
For O<i<N and j=O:

- [(m-i+l)A1Pl(i-l) - i~lPl(i)J/Pl(i)

+ [(m-i)A1Pl(i) - (i+l)lJ1Pl(i+l)]/Pl(i)

+ [nA2P2(0) - ~2P2(1)J/P2(O) = o.

For O<i<N and O<j<N-i:

- [ (m- i+1) A1PI (i-1) - i u1PI (i) ] / PI (i )

+ [(m-i)A1Pl(i) - (i+l)~lPl(i+l)J/Pl(i)

- [(n-j+l)A2P2(j-l) - j~2P2(j)J/P2(j)

+ [(n-j)A2P2(j) - (j+l)~2P2(j+l)}/P2(j) = o.

For O<i<N and j=N-i:

- [(m-i+l)A1Pl(i-l) - i~lPl(i)J/Pl(i)

- [(n-j+l)A2P2(j-l) - j~2P2(j)J/P2(j) = O.

For i=N and j=O:

(15)

The key point to observe in (15) is that all the sixteen square brackets are

structurally the same. One simply has for 0 i < Nand 0 j < N, a pair of

symmetric conditions:

(m-i)A1Pl(i) - (i+l)~lPl(i+l) = 0,

(n-j)A2P2(j) - (j+l)~2P2(j+l) = O.

(16 )

If these equations hold, one has a solution. The separated equations for

Pl(i) and P2(j) are both of the Engset type (Kleinrock, 1975, Chap. 3). Except for

the missing proportionality factor, their solutions are known~

29
(17 )

The resultant state probabilities are therefore

(18 )

(~)(j)a, i a2j

I (~)(~)a,sa2t'

all s!tt

p(i ,j) = ---------=----

where al=Al/~l and a2=A2/~2 are the individual loop and line loads in Erlangs!t respectively. The nontrivial domain for (i,j) is, of course, min{i,j,m-i,n-j,N-i-j}~O.

The state probability solution for Phase 2 differs from the approximate

solution for Phase 1,. Compare (5) and (18). However, their limiting _properties

are similar. Thus, for Phase 2 one gets,

if i =0,

- ........ 0,

al-+O

----.... E

i (N ;m,al ) ,

a2-+0 .

if i>O,

if j=O,

if j >0,

• m,n~

A iAj

1 2

-, -,

1 -J .

s t' L A, A2

all s,t sit!

all i, j, (19 )

where Al=ma l, and A2=na2 are assumed to be constants in the last limit. These

limits are formally the same as (7), (8), and (9) for the blocking problem in Phase

1. The last of the limits covers the case of both user groups being infinite.

This asymptotic result has been previously derived by Cooper (1972). For Phase 2

it implies the simplification,

30
k (Al+A2)

Pr{i+j=k} • N k! s' m,n~ L (A1+A2)

s=O sl

(20)

It leads to the Erlang B blocking formula, as noted later.

The general blocking probability for Phase 2 results when (18) is inserted

into (13). The P

x expression is a symmetric function of the two user groups~ If

one assumes N<min{m,n}, then probability P can be written as - x

N. al L (n,.)a2'F(-m,-i;n-i+l;--) i=O a2

(21 )

Here, the F(.,.;.;.) function is the Gauss hypergeometric series (Abramowitz and

Stegun, 1964).

F(i,j;k;x) (22)

where (i)n=i(i+1) ... (i+n-1) and (i)o=l. For negative i or j, the Gauss series

becomes a polynomial. That clearly is the case in (21).

4.3. Numerical Results

The numerical evaluation of blocking probability for the Phase 2 problem

follows from the formal solution, see equations (13) to (21). The task involves

five independent parameters: m, n, N, al and a2. Thus, like in earlier Phase 1,

it is the mechanics of presentation that causes difficulties, perhaps in excess of

the computational effort. In this section, a simplified summary of the numerical

results is presented.

As before, let Al=mal and A2=na2 be the total loads offereQ by the two user

classes. Let the total initial load to the service facility of Phase 2 be their

sum, A=Al+A2. One expects, quite reasonably, that said loads must playa key part

in determining the eventual grade of service. This parallels the rationale for

the Engset distribution, and the fact that -- at least marginally -- the present

solution coincides with the Engset formulas. See equations (19) and (20).

31
Special marginal cases for thePABX blocking probability are:

P

x

A = EN(N;n'n) if Al=O,

A if A2=O, = EN(N;m'm)

A if

Al A2 = EN(N;m+n'm+n) --

m n

lim A = EN(N ;k'k)=B(N,A) if both m=oo, n=oo.

k-+oo

(23)

The final form B(N,A) is the Erlang Bformula (Bear, 1976). The notation used in

(23) is the same as defined in (6).

The Engset distribution is known to possess several monotonic properties.

Thus, EN(N;k,~) decreases as N increases, and increases as either k or A increases

(Siemens, 1974). These properties can be variously extended and applied to the

Phase 2 solution. The typical characteristic of Figure 12 results.

Figure 12 shows the blocking probability, P , for Phase 2 as a function of the

x

relative total loop load AliA. This does not imply that the total load A is an

insignificant parameter. Quite the contrary, the magnitude of Pxis bounded and

determined by A, perhaps more than by anything else.

In the interpretation of Figure 12, assume that all ofm, n, N and A are fixed.

ThenAl is the only variable. Asa function of this variable, P

x obeys simple

monotonic rules. Its maximum value or worst case occurs when all loop and line

loads are the same. This happens when

(24)

At the other extreme, the function has two local minima. They occur at the end

points A1/A=O or 1, as shown. Between the maximum and the minima, the function has

a monotonic gradually increasing slope. The function is symmetric around the midpoint when m=n.

In access area applications, one expects fewer loops than lines, but with

higher loop traffic. Then m.~n and A1/A~/(m+n). It follows that the system

operation should be characterized by the right side of Figure 12, where

A . A

Er,,(N;m,-)<P <EN(N;m+n,.·-+). 'i m - x~ m n

32

(25)
w

w

CJ)

c

..x

U

o

CD

'+-

o

+-

..0

o

..0

e

Q..

P-4)(

1.--- I

I

-EN(N;n, ~)

EN (N; m+n, m~n )

( A, EN N; m, mJ

00 m

m-n

Relative Loop Load A,iA

Figure 12. The general effect of the Engset distribution on the

blocking probab i l i ty for Phase 2.

1
One should emphasize that two things happen as mand n are jointly increased.

First, the P

x curve is raised. And second, it is flattened out, eventually approaching the Erlang B(N.A) limit (Bene¥. 1965; Siemens. 1974; Kleinrock, 1975), that

was indicated in (23).

Several curves of interest have been computed. They are presented in three

parts of Figure 13. All the plots show P

x

as a function of the relative total loop

load AliA. All plots also ass~me twice as many lines as loops, i.e., m/n=1/2.

This implies~ see (24), that the maximum probability of blocking occurs at the

indicated A1/A=1/3. The maximum value is the worst case bound for given m, n, N

and A.

The three parts of Figure 13 differ in the N (i.e., number of servers) and A

(i.e., total initial load) values assumed. For other values, two choices are open

to the system analyst. One can perform exact calculations on the formal solution,

(13), (18) and (21), or one may resort to some, quick but rough, pseudo-parabolic

interpolation between the Engset extremes (25).

Further exploration of the complete parameter ranges for m, n, N, Aland A2

appears to be a sizable effort. It is beyond the scope of this paper.

5. BLOCKING PHASE 3

5.1. Problem Definition

Consider the service facility labeled II swi t chll in Figure 14. It is distinguished

from other previously analyzed facilities by the fact that it provides two kinds of

servers for three types of service requests (calls).

On the left side of the switch, there are a total of L channel ports, to be

loosely called "lines. 1I On the right side there are T IItrunks. 1I There are three

types of calls through the switch. The distinction is illustrated by the (a), (0)

and (E) paths in Figure 14. (Note that the distinction between previous (a) and

(y) calls becomes blurred here.) For lack of better names, the (a)-type traffic

will be referred to as IIdistant;1I the (o)-type as 1I1 ocal;1I and the (E)-type as

IItandem. 1I The route profiles of Table 1 again show that many paths belong to

these categories. For instance, all the paths that go through switches more than

once, such as 1, 2, ... , 27, 28, belong to type (a) at Phase 3. If a switch is

encountered once in Table 1, as for paths 5,6, ... ,31,32, then one has category

(0). Finally~ of course, every member path of (a) may well be a member of (E)

class" somewhere down the route.

34
-2 5X10' ii' , 5X10-2

m=24

2

2

5

10-2

m=a>

m=6

IN=6 I 2 A=2.4

2

Px

W

c..;l

Figure 13. Phase 2 blocking probabilities for loop-to-line ratio nl/n=1/2.
L. iT

PABX SWITCH

I CONCEN- I I

~I TRATOR w

C)

\ , v

Blocking

Phase 3

figure 14. Phase 3 - blocking at a switch.
This section derives blocking probabilities for all three kinds of calls.

The probabilities are inherently different, because distant calls require one idle

line plus one idle trunk, local calls require two idle lines, and tandem calls need

tVJO idle trunks.

It is recognized that blocking for Phase 3 is a generalization of Phase 1

(see Section 3) and therefore apparently untractable. However, after invoking a

few reasonable assumptions, the problem becomes simpler. That is done in this

section. Formal closed form expressions are next derived for both the state and

blocking probabilities subject to simplifying assumptions.

The simplifying assumptions are:

(i) The interarrival time distributions for the three kinds of service

requests are exponential, all possibly distinct. The service times are

also exponentially distributed, but with a common mean of unity. Their

Erlang loads may differ.

(ii) All service requests are generated by infinite user populations. This

assumption appears reasonable for switches that serve several concentrators,

several PABX's, and a largenurnber of terrninals. The three arrival

intensities thus stay constant, regardless of system state.

(iii) The number of servers, i.e., the L lines and T trunks of Figure 14, are

given constants.

(iv) All blocked calls are lost without any aftereffect to the system.

(v) The switching network is intrinsically nonblocking. By that one means

that blocking events can only be caused by a shortage of lines, trunks

or both.

(vi) The switching network provides full availability from all requesting

lines and trunks, to all idle lines and trunks on the network.

(vii) The network model in Figure 14 is equivalent to two (i.e., k=2) channel

groups with nl=L and n2=T servers, respectively. Amore representative

model should permit k~2 channel groups with {n l, n2, ... , nk} servers

each. The total number of traffic streams would then amount to

k(k+l)/2, instead of the present three. Solely for reasons of simplicity, the general model is not pursued here. One suspects, nevertheless,

that the solution for general k is tractable and in fact similar to

k=2 in structure. This general topic should be best ~xplored in a

separate in-depth study.

37
To be specific, assume the notation summarized in Table 5. When d distant

calls, £ local calls, and t tandem calls are in progress, one says that the system

is in state (d,£,t). The steady state probability of state (d,£,t) is p(d,£,t).

Quantities~ P

d, P£ and P

t, denote the probabilities of blocking for the distant,

local and tandem calls, respectively. The total offered loads for the three

traffic types will be A, Band C Erlangs. To obtain se-rvice, a distant call

requires that at least one line and trunk is idle. As shown, local calls need two

lines and tandem calls need two trunks.

Nonzero probabilities can occur only when min{d,£,t,L-d-2£,T-d-2t}~O is true.

Impossible (d,£,t) states have p(d,£,t)=O, by definition. The blocking probabilities are sums of p(d,£,t) over appropriate index sets (d,£,t). To this end, one

next derives the individual state probabilities~

The derivation of the p(d,£,t) formula~ to be given next, parallels the procedure for blocking Phases 1 and 2. One constructs and solves the steady-state

flow conservation equations for unknowns p(d,£,t). As before [see (4)] two column

vectors are involved. First, let the following seven-dimensional vector E represent the unknown state probabilities,

p(d-l,£,t)

p(d,£-l,t)

p(d,£,t-l)

P = p(d,£,t)

p(d+l,£,t)

p(d,£+l,t)

p(d,£,t+l)

(26)

The second seven-dimensional vector, f, is comprised of the flow rate or

birth-death coefficients necessary for flow conservation at every state (d,£,t).

Because of the network complexity, one must now distinguish at least seven separate

(d,£,t) regions, each with its own coefficients. These seven state regions are

defined in Table 6. Each region (k=lt2, ... t7) has a unique coefficient vector~.

The blocking events, be they for distant, local or tandem calls, are also different. The blocking occurrences are indicated in Table 6. Note that region k=l

blocks all calls. Region k=7 blocks no calls. The other regionst

, 2 to 6, have

different individual blocking characteristics.

38
Table 5. Notation for the Three Kinds of Service Requests

from Two Classes of Server Channels

Requ i red Number

Type of State Probe of Offered of Idle

Traffic Index Blocking Load

Lines Trunks

Distant

(a) (d,.,.) P

d A 1 1

Local

(8) (.,~,.) P~ B 2 0

Tandem

(E:) (.,.,t) P

t C 0 2

39
Table 6. The Seven State Regions and Their Blocking Roles

d+2t=T d+2t=T-1 d+2t<T-1

k=l Distant k=4 Distant

Local d+2~=L Local --

d+2£=L-l Tandem k=2 -- k=5 -- Local Local

Tandem --

k=3 Distant k=6 -- k=7 -- d+2£<L- 1 -- -- _..- Tandem Tandem --

40
The seven-dimensional column vectors are given in Table 7, for all seven

regions k=1,2, ... ,7. For reasons of simplicity, just like in Tables 2 and 3 the

trivial min{d,£,t}=O cases are avoided here. The flow conservation identities,

see earlier equation (4), consist of products of all (i.e., the columns of

Table 7) with f (26). The solution must again be normalized,

I p(d,£,t) = 1.

all d,£,t

(27)

5.2. Formal Solution

As for Phase 2, one tries a separable or product solution (Cooper, 1972),

(28)

The substitution of (28) into (26), followed by a second substitution of (26)

and Table 7 into (4), produces a long string of vanishing sums. But, just as in

(15) and (16), the sums are very simply constituted. They all turn out to be

variously weighted sums of three elementary equations. The elementary equations

are

dp, (d) AP1(d-l) = 0,

£P2(£) BP2(£-1) = 0, (29)

tP3(t) CP3(t-l) = o.

They are easily solved. After all, identities of type (29) have been the basis

for the classical Erlang formulas (Riordan, 1962, Chap. 5; Kleinrock, 1975, Chap. 3).

The separated solutions are thus known to within a multiplicative constant:

Pl (d)

Ad a: -d! '

P2(£)

B£ a: - (30) £1 '

P3(t) C

t a: -t! ·

Equations (27), (28) and (30) uniquely determine the state probabilities for

the Phase 3 blocking problem. For all nonnegative d, £, and t values that satisfy

d+2£~L and d+2t~T, one obtains

41
N

Table 7. The Column Vectors for the Seven State Regions k=1,2, ... ,7

-----,

~l ~2

, f 3 I I f I c

i ! I -7 I

I

-A -A . i -A -A -A -A -A

-8 -8 -8 -8 -8 -B -B

-C -c -C -C -c -c -C

d+£+t d+£+t+A d+£+t+B d+£+t+C d+£+t+A+C d+£+t+A+B d+£+t+A+B+C

0 -(d+l) 0 I 0 -(d+l) -(d+l) -(d+l)

0 0 -(£+1) 0 0 -(£+1) -(£+1 )

0 0 0 -(t+l) -(t+l) o I -(t+l) I
(31 )

AdS£Ct

d!£!t!

L AiBjCk ·

all i,j,k i!j!k!

p(d , £ , t ) = -------::~-::--

Three distinct blocking probabilities are encountered here. See Tables 5

and 6 for notation. The probability of distant call blocking, Pd, is the sum of

all state probabilities in the index set k=1,3,4. The probability of local call

blocking, P£, is aggregated over the index set k=1,2,4,5. And, finally, the

tandem call blocking, P

t, comes from k=1,2,3,6.

5.3. Numerical Results

Expressions for three kinds of blocking probabilities, P

d, P£, and Pt , have

been formally derived. It remains to evaluate them numerically for the appropriate

parameters. The three probabili~ies of blocking, P

d, P£, and Pt, are functions of

five independent parameters: L, T, A, B, and C. The computation and presentation

of results appears again to be rather involved.

In Figure 15 one finds typical and slightly oversimplified numerical results.

Figure 15 has three parts. All show the three blocking probabilities as ordinates,

versus the tandem call load C as abscissa. Furthermore, all three parts assume

that the number of lines and trunks are L=12 and T=6, respectively. The three

parts differ in the distant (A) and local (B) call loads. The first part depicts

A=O.5~ B=l Erlangs. The second has A=O.75 and B=1.5, and the third A=l, B=2

Erlangs. Thus~ all nine curves of Figure 15 correspond to fixed ratios L/T=B/A=2.

The tandem load C is seen to have a marked effect on the tandem call blocking

probability, Pt· At C=O, P

t has its lowest value. As C increases, P

t grows in a

monotonic fashion. A somewhat lower rate of increase is encountered by Pd , the

probability of blocking for distant calls. With respect to C, P

d also has a

minimum at C=O~ At that point, P

d is a special case of the distant call blocking

Po' discussed under Phase 1 blocking (see equations' (2) to (11), and Figures

4 to 9). In the cases considered so far, one finds Pd>Pt at C=O. However, because

of their different rates of increase as functions of load C, the roles are soon

reversed for relatively modest C>O. One finds in Figure 15, that Pd<Pt for all

C>0.12 Erlangs ..

Blocking probability for local calls, P£, is nearly independent of C for the

cases considered. This is expected. However, there is a slight yet discernible

paradox here# The highest value of P£ occurs at C=O, as can be noted from a careful scrutiny of the curves. Increases of C~ which happen to block trunks needed for

distant traffic, may occasionally augment the idle available lines.

43
+~

g 101

u

o

CD

'0

.:t:

:0

o

..c

et 102

L=12

T=6

A=0.5

B =1

L=12

T=6

A= 0.75

8=1.5

6 0.8 1.0 0 0.2 0.4

L=12

T=6

A=l

8=2

0.6

Pt

l}

10-1

10-2

Figure 15. The three blocking probabilities of Phase 3.
Figure 15 is an initial step towards full Phase 3 representation of a

military access area switch. Larger sizes and more channel groups must be addressed,

as mentioned earlier. The number of lines, L, and the number of trunks, T, are

usually in hundreds, if not thousands. The loads, offered and carried, must also

be magnified accordingly. Unfortunately, however, the computation of the five

parameter (i.e., L, T, A, B, C) effects on the three blocking probabilities (i.e.~

P

d, P£, P

t) is already quite complex. A more comprehensive numerical effort may

be warranted, but was not undertaken as part of this study.

A simplified symmetric case has been evaluated. It assumes that the local

and tandem loads are the same, and so are the number of lines and trunks provided.

An example is given in Figure 16. It postulates modest offered loads B=C=5 Erlangs

on both sides of the switch. One also assumes that a T1 line, or its equivalent

24-channel set, serves each side of the switching network. As abscissa, Figure 16

varies the distant call load A. Because of the inherent symmetries (i.e., L=T and

B=C), the local and tandem call blocking probabilities are identical in the figure.

The second curve depicts the distant call blocking probability.

When A=O, the Erlang B formula (Cooper, 1972; Bear, 1976) applies. Then

(32)

However, as A increases, the roles of Pd and P£=Pt become reversed. While P£=Pt

increases in a monotonic way, the distant call blocking probability, P

d, first

decreases~ passes through a minimum, and finally also increases as a function of

load A. For the symmetric case L=T=24 and B=C=5 Erlangs, P£=Pt>Pd holds for A>O.5

Er1angs.

6. APPLICATIONS OF THE THREE BLOCKING RESULTS

6.1. Interpretation and Combination of Results

For the purpose of estimating access area grade of service, previous sections

have structured the access area network operation in a special way. At key nodes,

the communication paths are partitioned in any of three ways. These ways are

called Phases 1, 2 and 3. Probabilities of blocking are derived, and to a limited

extent, computed and graphed for these phases.

A total of six blocking probabilities have been identified in Sections 3, 4

and 5. A given network path can encounter various sequences of these six blocking

events. Early in this report, Table 1 outlined forty potential route profiles for

the access area teletraffic. Everyone of said routes passes either zero times,

45
L = T =24

B =C = 5

CJl

C

.x:

U

o

m

o 246 8

Distant Load A [Erlangs]

10

Ii qure 16" Phase 3 bl ockinq 'for a synmetric switch (L=T) with

a synunetric load (B=C) ..

46
or once, or twice through the blocking phases. In doing so, the routes encounter

blocking possibilities of the various types, and with different multiplicity per

each path. Table 8 brings this all together. It shows for each of the forty

paths, the incidences (i.e., the number of occurrences) of the six blocking elements.

The header of Table 8 identifies the three phases of blocking. Phase 1,

which takes place at a concentrator, offers two probabilities of blocking:

Po - for outgoing/incoming through-traffic,

P

r - for return traffic to the same concentrator.

As seen from Table 8, most of the forty paths encounter Po either once or

twice. On the other hand, Pr

occurs only for one path, namely for #36. The

properties and numerical values for both P and P are presented in Section 3. o r

They both depend on:

M- the number of concentrated user terminals,

m - the server channels,

a - the out/in load per terminal in Erlangs,

b the return load per terminal, also in Erlangs.

Of the two probabilities, as one expects, P is the largest. r

Phase 2 blocking occurs at a PABX. Its probability is denoted in Table 8 by

Px. One notes that the majority of the forty path types face P

x

either once or

twice. Section 4 presents the P material. It depends on such parameters as:

x

m - the number of concentrated loops,

n - the number of lines,

N - the number of server channels,

al - the offered load per loop in Erlangs,

a2 - the offered load per line in Erlangs.

The header of Table 8 shows that blocking at Phase 3 can appear in three forms.

First, there is the distant call probability of blocking, P

d" It occurs in pairs,

on 16 out of the 40 paths. Next, the local call blocking, P£, occurs only in

singles, but on another set of 16 path types. Finally, there is the tandem call

blocking probability, Pt. While not of immediate interest to the access area

47
Table 8. Blocking Probability Structures for the Forty Access

Area Circuit Paths

Blocking Probability Elements

Phase 1 Phase 2 Phase 3

Path

# Po P

r

P P

d x I P,Q, rPt

(Out/In) (Return) (PABX) (Distant) (Local) (Tandem)

1 2 2 2 ... 2 1 2 2 ·.. 3 2 1 2 ... 4 1 1 2 ·.. 5 2 2 1 j

6 1 2 1

7 2 1 1

8 1 1 1

9 1 2 2 ·.. 10 2 2 ·.. 11 1 1 2 ... 12 1 2 ·.. 13 1 2 1

14 2 1

15 1 1 1

16 1 1

17 2 1 2 ·.. 18 1 1 2 ... 19 2 2 ... 20 1 2 ... 21 2 1 1

22 1 1 1

23 2 1

24 1 1

25 1 1 2 ... 26 1 2 ... 27 1 2 ... 28 2 ... 29 1 1 1

30 1 1

31 1 1

32 1

33 2 2

34 1 2

1

35 2 1

36 1

I 37 1 2

I 38 2 1 39 1 1

140 1 ;. I

48
study, P

t seems to have potential of occurring any number of 0,1,2, ... , times for

all those 16 paths that encounter P

d twice.

The three Phase 3 blocking probabilities, P

d, P~ and Pt, depend on:

L - the number of serving lines,

T - the number of serving trunks,

A the total offered distant load in Erlangs,

B - the total offered local load in Erlangs,

C - the total offered tandem load in Erlangs.

The derivation of Phase 3 formulas, as well as the related P

d, P~ and Pt curves,

is presented in Section 5.

Table 8 offers a prescription on how to add the various blocking probabilities

for various teletraffic profiles. We shall demonstrate this later by application

to specific access area implementation.

6.1.1. Worst and best cases

A glance at Table 8 tells that certain paths· blocking probabilities are

bounded by others. Some are simple ordered upper bounds and others are lower

bounds. There are also cases where direct comparison is impossible without

additional information.

Let P(k) be the effective access area end-to-end blocking probability for

path k(k=l ~2, ... ,40). As is appropriate for access area, exclude the tandem

blocking, P

t, from this total. If the blocking events are roughly independent,

P(k) must be approximately equal to the sum of the probabilities listed in Table 8

for path k. Under such a premise a partial ordering of P(k)·s is possible.

Without going into details, one realizes that the highest probability of blocking

must occur on either path 1, 5, or 36. The least amount of blocking must occur

on either path 28, 32, 36, or 40.

Formally, the worst and best case bounds must therefore be

P(k) max{P(l), P(5), P(36)}

min{P(28), P(32), P(36), P(40)}. (33)

And if the individual probabilities are realistically small, additive, and consistent,

then

49
P(k) max{Pr, 2Po + 2Px

+ max(2Pd, Pi)}

min{Pr, Px' 2Pd, P£} (34)

holds for all access area paths listed in Table 8, k=1,2, ... ,40. (Note: Po is

said to be consistent or uniform in an access area, if it has roughly the same

value throughout that area. The same consistency definition holds for P

r , Px'

P

d , and P£, as well.)

The first line of (34) is suited for conservative access area end-to-end

grade of service estimation. Thus, by omitting tandem blocking effects at switches,

it is quite easy to estimate the worst case bound for blocking on all forty paths.

Examples:

If Po = Px = P£ = 0.002,

P

r = 0.010,

P

d = 0.001,

then the overall blocking cannot exceed 0.01 or 1%.

If Po = Pd = 0.010,

P

r = 0.050,

P

x = 0.005,

P£ = 0.020,

then the probability-of blocking must be lower than or equal to 5%.

6.2. Specific Application Examples

Teletraffic engineering parameters developed i~ the previous sections have

bearing on the design of new, and upgrading of existing, networks for military

access areas. The relationships between the parameters tell one whether the sizes

or structures of certain network elements meet the grade of service requirements.

This section demonstrates the procedure. The teletraffic profile of an

unspecified access area is used to test the implementation. The profile is ~pical of a large access area. It contains a main post, plus several somewhat isolated

camps and military stations nearby. The implementation concept is perhaps unique.

Digital technology is used for concentration, switching and transmission. Both

analog and digital services may be integrated on the network.

50
6.2.1. Access area and its communications outline

The posts, camps and military stations of an access area share the local telecommunication facilities. Many existing areas include tactical units, military

reservations, and other entities whose communications systems planning and implementation is the responsibility of a single military department. To overview them

briefly, note the following.

There are approximately 1500 military access areas worldwide. Based on the

number of user terminals in each area, the areas range from small «300 terminals),

to medium (300 to 3000 terminals), to large (>3000 terminals). The majority of

terminal types are telephones, but they also include computer, teletypewriters,

facsimile, and a myriad of other terminals. Terminal densities vary from less

than 10 per km2 to over 10,000 per km2. The higher density cases offer a variety

of line concentration alternatives.

Table 9 summarizes the military communications environment. In addition to

sizes and population estimates, Table 9 lists the number of various communications

terminals which might be found on a medium size post; in an entire relatively

large access area; a large military region (consisting of multiple access areas);

the continental United States (CONUS); and the global arena of the entire outside

world (OCONUS).

Local access area traffic destined for other areas uses the military long haul

transmission facilities. There are interfaces between the access area networks

and the long haul networks (see Figure 1). The future long haul facilities may

interconnect local access areas on either a smaller or larger area basis by a

hierarchical network structure. A possible structure is illustrated in Figure

17. It shows that terminals in an office complex first access a remote switch

hub, or PABX, in the office complex. Through that PABX they gain access to the

central switch hub or the dial central office on the base, followed by the regional

hub or access area switch, and finally to the backbone switch on the Defense

Communication Systems (DCS) Global Network.

At the same time, each local area may access common carriers for domestic

services. While not highlighted in Figure 17, this fact was illustrated earlier

in Figure 1.

Let us return to a typical, medium size, local access area. As stated, it

consists of a main post, plus several camps and military stations within its

immediate environs, some 1 to 20 km away. Let one station be a large office and

laboratory complex outside the main postls reservation boundary.

51
Table 9. Estimates of Military Communications Terminals,

Local to Worldwide

Medium Access ~1i 1i tary CONUS OCONUS Post Area Region

2 Area (km ) 5 50 5,000 15M 500M

(Worldwide)

Population

(Military & civilian) 5,000 15,000 50,000 1. 5~1 o. 5~1

Telephones

Mainline 1,500 6,000 15,000 300,000 50,000

Extensions 1,000 4,000_ 10,000 200,000 30,000

Sub-totals 2,500 10,000 25,000 500,000 80,000

Data Terminals

Interactive 10 30 100 1,000 400

Computer 2 6 30 200 50

Narrative 5 15 50 500 100

Facsimi1e 3 8 25 250 50

Data 2 6 30 300 50 - - - -

Sub-totals 22 65 235 2,250 650

Secure Terminals

AUTOSEVOCOM 1 3 10 1,000 500

! I

52
To other

Backbone Nodes

Main Access

Area Switch

Central

Switch Hub

Remote

Switch Hub

Communications

Terminal

Backbone Switch

Main

Access Area

Access

Area Segment

Figure 17. Hierarchical network configurations for interregional

access.

53
A telephone terminal density profile for this illustrative area is postulated

in Table 10. Each post, camp and station is divided into subareas corresponding

to the average neighborhood terminal densities. This division determines the

numbers and the locations of terminal clusters, concentratitinpoints, remote

switching hubs (e.g., PABX·s) and central offices.

A further result from the telephone density breakout is the estimate of the

number of~ainlinesand bridged lines required to serve the area. Table 11 lists

such a telephone dispersal versus eight subarea types, from high density offices

to low density suburban. Since the distinction between various office and residential service areas tends to be subjective, the ratios of mainlines to total terminal

number are always somewhat arbitrary.

6.2.2. Network implementation concept

Next, one must configure the network. An implementation concept for an alldigital access area network is shown'in Figure 18. Terminals interface with

switching hubs via concentrators whose outputs are time division multiplex (TOM)

loops. Concentrators may be remoted from the switches themselves to sites at or

near the terminal clusters. Repeatered trunk circuits are used for such remoted

units whenever the span ;s sufficiently large. Trunk circuits may be multiplexed

to higher MUXleve1s, to relatively standard 24 channels (Tl), or 48 channels

(T1C), or 96 channels (T2) -- all digital trunks.

The remote switch hubs (RSH), such as PABX·s, are located in office complexes.

Local concentration units (LCU) provide terminations at the switch itself. Remote

concentration units (RCU) are some distance from the RSH, at strategic clusters of

terminals.

The switching is performed by time slot interchange on the concentrator

output loops, or by time multiplexed gates on space division multiplexed (SDM)

highways (Linfield, and Nesenbergs, 1978).

The local and remoted components of the swftching system are shown in Figure

19. The size of the switch is partly determined by the RCU·s and LCU·s that home

on the switch (Linfield and Nesenbergs, 1979). At the design level, the concentration ratio depends on the projected traffic intensity.

All inputs and outputs of the switching matrix are assumed to consist of

multiplexed l2ttme slot units. A time slot contains one 8-bit byte of a standard

64 kb/s pulse code modulated (PCM) signal. Switching occurs 8000 times per second

for each such voice-equivalent channel.

54
(J1

(J1

Table 10. Telephone Terminal Density Profile

Subarea DensitY2 Subarea in km 2

Total

Type Term/km ~1a in Post Remote Camps Remote Stations Total Area Terminals

Large Office 10,000 0.15 0.2 0.3 0.65 6,500

Medium Office 5,000 0.2 0.2 -- 0.40 2,000

Small Office 2,000 0.4 -- 0.02 0.42 820

Barracks 1,000 0.15 -- -- 0.15 150

High Residential 500 0.5 0.4 -- 0.90 450

Low Residential 200 0.2 0.1 -- 0.30 60

High Suburban 100 0.2 0.05 -- 0.25 25

Low Suburban 50 0.1 -- -- 0.10 5

Total Terminals 3765 3225 3040 10,000

; !
U1

m

Table 11. Main Line and Bridged Telephone Distribution Profile

Main Post Remote Camps Remote Stations

Subarea ~1a i n Lines I r~ain Lines Main Lines

Type + Bridged Main Line + Bridged r

v1ain Line + Bridged tt1a in Line

Large Office 1,500 700 2,000 1,300 3,000 1,600~

~1edium Office 1,000 400 1,000 600 -- -- -

Small Office 800 300 -- -- 40 25

Barracks 150 I 80 -- -- -- --

I High Residential 250 90 200 140 -- --

Low Res i dent i a1 ' 40 30 20 10 -- --

High Suburban 20 20 5 5 I -- --

I

Low Suburban 5 5 -- -- -- --

Totals 3,765 1,625 3,225 2,065 3,040 1,625

!
Line Side

Subscribers

Terminal

/

v

Central Remote

Switching Concentration

Hub Hub

Repeatered

Trunk

v

Remote

Switching

Hubs

Line Side I Trunk Side

I

I Repeatered I

Line I

I

I

" v

Remote

Concentration

Hubs

Ff qure 18" Implementation concept for digital concentration,

multiplexing, switching and transmission in the

access area"

57
Space --I

Switch

~Remoted Components -------::II.....__~-Switching Hub --------.....

TMG

DT

DVT

r---,

I I

I

I

I I L __ -,

TIU

r--,

I I ....-J--------1-f ">-I----~~-----~---:-----.

T-Trunk I I

~-----~---""-"-"'------------t:tt----t:tt--t:II I I L __.J

TRU

Legend

DSVT: Digital Secure Voice Terminal

AVT: Analog Voice Terminal

DT: Digital Terminal

DVT: Digital Voice Terminal

LCU: Local Concentration Unit

RCU: Remote Concentration Unit

TRU: Trunk Repeater Unit

TIU: Trunk Interface Unit

TSI: Time Slot Interchange

TMG: Time Multiplexed Gate

Fi qure 19. Local and remo ted components of a sw itch inq hub.

58
The above implementation concept has been applied to the access area with

population, terminal and mainline density profiles given in Tables 9, 10, and 11,

respectively. The result is the topological structure of Figure 20. In this

figure, the numbers next to a terminal symbol denote the number of such terminals

that are concentrated at the next higher stage, such as at RCU, LCU, RSH, and so

forth.

The remaining traffic engineering job is to ascertain the blocking probability

levels for the family of forty potential paths (see Tables 1 and 8), as they traverse the network of Figure 20 in all conceivable directions. Note that two of

the forty paths, namely #5 and #36, are drawn with broken lines in Figure 20. The

following paragraphs of this section estimate the blocking probabilities. The key

tools will be the five blocking probabilitie~ (i.e., see Table 8 and ignore Pt),

that result from the three phases of access area traffic flow (see Sections 2, 3,

4 and 5).

6.2.3. Grade of service estimates

In this section, the estimation of numerical values is demonstrated. The

example applies to the network of Figure 20, and the various paths through it (see

Table 1). The blocking probabilities are determined in terms of the three blocking phases.

First, however, one additional piece of information is needed. That is the

amount of traffic offered and/or carried by the scenarios of Tables 9, 10, 11, and

Figure 20. For simplicity, assume that the loss is so small as to be negligible

to the gross flow. Then the carried load is practically indistinguishable from

the offered load, and one can refer to them both as the "load. 1I As before, let

all loads be expressed in Erlangs.

To expedite matters, assume the average loads per line to be as given in

Table 12. There are three number columns in this table. The first column shows

how the traffic from a station that is tied to a concentrator is distributed to

six different terminal groups. Finer resolutions of the substreams are possible,

but not attempted here. The key point is that such a concentrated station carries

0.1 Erlangs, of which half returns through the same CONC.

The second column depicts an access area terminal that is tied directly to

the PABX. Its load, perhaps as an anomaly, is shown to be less than that of the

first column. Finally, the third column applies to a station that has a direct

local loop to the end office switch. Its load is typically the largest and, as

59
55

90

300

300

300

100

200

500

RCU's

Main Post

Low Res. a Sub

HighResidential

\

\

Common Carrier

Des Backbone

Small Remote

Office

15

Camp Residences

Path#36

25

Remote Camps

and Stations

\

\

140

400

600

300~

1200

3690'IV Mainline

U Total

Figure 20. Illustrative example of an access area communications

network.

60
Table 12. Assumed Loads Per Line (In Erlangs) for Selected

Terminal-to-Terminal Group Substreams

From a Terminal that is Directly

Homed to

0..- CONC PABX SWITCH

Same Same Same .050 .022 .042

None, ,

C'O Same ..s::a

-I-J~ or

0..-0 Diff Diff Same .022 .010 .042 ::sClJ I a E s, a

I (..!J ::c

None .010 r-~ Same Same .008 .017 C'O rC-I-J

e,.... U

s,

EQJs, None Diff Same .005 -.006 .017

QJe,....

r- 0

C'O(/) e,.... None None Same .010 .008 .025

a

rNone None

or or

Diff Diff Diff .003 .006 .017

Total Per Line Load at a

Terminal .100 .060 .160

61
seen from Table 12, it is assumed in this sample to be equal to the sum of the

previous two.

Incidentally, the general distribution of per line intensities is roughly

comparable in the three columns. All substreams, and there are 18 such in Table

12, are identifiable in the access area network (Figure 20), with groupings of

circuit paths (Table 1), as well as with elements of the three-phase blocking

model (Figure 2).

The quantitative information provided about the access area teletraffic, plus

the three-phase approach to blocking, suffices for estimating grade of service for

any given service line and trunk group provision. That means that any network

configuration, such as that given in Figure 20, may be specified in terms of

blocking probabilities P(k) for all forty paths k=1,2, ... ,40 (see Tables 1 and

8) A

To show one way of estimating grade of service quickly, assume that all

network elements in the present access area example (Figure 20) are grouped in

modules equivalent to or better than the module in Figure 21. The concentrators

compress M=96 user lines into m=24 CONe loops. These, in turn, are grouped with

n=48 direct lines at PABX line side. On the other side of the PABX, a tie-line

group of N=12 provides access to the central switch hub. Here, the switch is

depicted as a rather uncommonly small one. The choice of L=T=24 (lines and trunks,

respectively) is strictly for illustration sake. It enables one to get along with

the very limited graphs computed in Section 5. (For most switches, with larger L

and T groups, additional computation of P

d, Pi/, and P

t formulas appears necessary.)

Table 12 establishes the loads at the concentrator:

M= 96,

m = 24,

M(a+b) = 9.6 Erlangs,

a

a+b = 0.5,

and thus from Figure 7, the out/in and return call blocking probabilities are

Po :: .005,

Pr :: · 010. (Phase 1)

62
I CONCEN- I I -----II TRATOR

T=24

Trunk

Group

SWITCH

L=24

PABX

1

I

I . I I

I N=12 I

Tie-Line l

Group L-N=12

Line

Group

I

I

I

m=24 1

Cone I

Loops n=48

Lines

f

I

M=96

Lines

m

w

Figure 21. Sizing of 1ine and trunk groups in the three phase f ormat .
At the PABX:

m = 24,

n = 48,

Al = 4.7 Erlangs,

A2 = 2.5 Erlangs.

From Figure 13 then, the PABX blocking loss is

P

x

.012

Finally, at the switch:

L = T = 24,

(Phase 2)

A = 1.6 Erlangs,

B = C = 5.0 Erlangs.

Here Figure 16 applies, and the distant, local and tandem call probabilities are

appoximately

Pd = .008,

P£ = Pt .010. (Phase 3)

The probabilities of blocking for the above phases must be combined to estimate the end-to-end blocking for the forty circuit paths (see Tables 1 and 8).

When Po' P

x

or Pd occur twice, one assumes that multiplication by two is justified

in view of the uniform modular layout (Figure 21). The individual Po' Pr' ... '

probabilities have all been shown above to be of the order of 1%. Hence intersections of second and higher orders, as well as statistical correlations can be

realistically neglected. Sums of the appropriate multiples of the element probabilities provide an adequate total probability estimate.

Table 13 is the result. It lists the blocking probabilities for all forty

paths under the above access area assumptions made. In particular, this table

reflects the implementation of the illustrative network (Figure 20) with the

concentrator-PABX-switch modules of Figure 21, or their equivalents. The table

does ignore, as stated earlier", all tandem blocking in the long haul networks.

Table 13 reveals a five-to-one range in P(k) values. The largest value of 5%

loss is seen for path #1. The least value of loss is 1%. It arises for two paths,

#32 and #36. These extremes agree with the ranges described in equations (33) and

64
Table 13. Blocking Probabilities for All Forty

Paths for the Assumed Access Area

Path Blocking

Number Probability

k P(k)

1 .050 (max)

2 .045

3 .038

4 .033

5 .044

6 .039

7 . 032

8 .027

9 .045

10 .040

11 .033

12 .028

13 .039

14 .034

15 .027

16 .022

17 .038

18 .033

19 .026

20 .021

21 .032

22 .027

23 .020

24 .015

25 .033

26 .028

27 .021

28 .016

29 .027

30 .022

31 .015

32 .010 (min)

33 .034

34 .029

35 .022

36 .010 (min)

37 .029

38 .024

39 .017

40 .012

\ .

65
(34). Of the two, the largest represents the worst overall case. It appears more

useful in access area teletrafficengineering.

Amean, f, averaged with the relative weights of the appropriate substreams

(as summarized in Table 12) is straightforward to calculate. One gets P=.0408, or

roughly a 4% loss, for the access area model assumed.

7. CONCLUSION

Tools have been developed for analysis of military access area circuit switching grade of service. The tools consist of structuring the network flow into

three distinct congestion phases. These phases (see Sections 3, 4, and 5) are

individually quite distinct. From a switching point of view, they represent new

problems that warrant new solutions.

This report has derived solutions for all three phases. The Phase 1 solution

is perhaps most succinctly stated in equation (10). It is a linear weighted sum

of two apparently reasonable approximations (5). The latter also provide valid

bounds. As spelled out in text, the solution for Phase 1 i-s exact for many

parametric ~onditions of interest. Unfortunately, the general role of the Phase

solution is that of an approximation. Its goodness of fit has not been formally

established, but is expected to be quite sufficient for access area applications.

The goodness of fit could be a topic for further work. Another extension of

Phase 1 should address the existing multiplicity of loads. Equation (12), or its

equi~alent, can be used to map initial loads into offered, as well as into effective or measured, loads at the concentrator facility.

The general state probability solution for Phase 2 is given in equation (18).

Itis an exact result for two user classes. Extensions to more user classes at

a PABX may be of interest. If so, they could be addressed in future studies. But

a far more significant generalization than the number of classes seems to be the

SENET type generalization. As mentioned in this report and reviewed elsewhere

(Vena and Coviello, 1975; Weinstein, et al .,1978; Blackman 1979; Gallagher~

1979), integrated voice and data, AUTOVON and AUTODIN, message and packet, transmission and switching, rerouting and queueing, plus other complex issues, enter

into the DCS transition thrust from the present analog toward the all-digital

system of the1990·s. The Phase 2 model should accordingly be extended to include

priority structures and pre-emption interrupts. This may alter the state and

blocking probabilities to a ~onsiderable degree. One should also incorporate

store-and~forward queueing aspects. In addition to the hybrid formulas for blocking (Nesenbergs, 1979), this elicits a statistical framework for random delays.

66
Such random delays are apt to occur at all storage nodes in the network (Baran,

1964; Kleinrock, 1976; Fischer and Harris, 1976), in a statistically dependent

manner, and with peculiar joint distributions. Future studies of local access

areas should take a close look at the priority delays generated by digital message

or packet switched nodes. After all, that may be the future function of access

area switching hubs.

The formal solution for Phase 3 state probabilities is given in (31). The

formula is exact for the assumptions stated. Representing a switch, the model

merits all the comments and extensions directed to the Phase 2 PABX. That, of

cour~e, includes storage delays and priority aspects. But, more significantly, a

larger number of channel groups - instead of the two treated in text - should be

analyzed. From all preliminary indications, this task appears tractable whenever

an infinite number of traffic sources may be postulated. Other issues at the

switch should" address limited or partial availability, as well as effects induced

by control and network structures.

Blocking probability curves have been computed for selected cases. While

these cases have turned out to be quite useful here (see Section 6) to demonstrate

the applications, additional computations seem needed for more realistic parameter

ranges of interest.

Both in the early parts (Sections 1 and 2) and in the later part (Section 6),

the definitions and the parametric features of the military access areas have been

emphasized. This appears natural, as the three phase approach has been motivated

to a large extent by traffic engineering in the local access area. This does not

mean, however, thatgeneralizations and extensions to other areas of military or

civilian teletraffic are impossible. The opposite seems to be true. As indicated

in the description of Phase 2 (see Figure 11, for instance), the latter can be

applied to several situations outside the access area proper. It is hoped that

all three blocking phases and their formulas, as well as their potential future

extensions - perhaps in ways not immediately obvious to these writers - may be

useful to communications network and traffic planners.

8. ACKNOWLEDGMENTS

The authors are indebted to many individuals either for their direct or indirect

support of this work. The list starts with Mr. T. Michelli, U.S. Army Communications

Systems Agency at Ft. Monmouth, NJ, who initiated and persisted in raising many of

the crucial questionse The Associate Director of ITS, Dr. P. McManamon, provided

the right environment. And finally, Mr. M.J. Miles of ITS offered computer programming and numerical resource help beyond all expectations.

67
9. REFERENCES

Abremowi tz ; M., and I. A. Stegun, Eds. (1964), Handbook of Mathematical Functions,

(National Bureau of Standards, Washington, DC), Applied Math. Series, No. 55.

Arthurs, E., and B. W. Stuck (1979), A theoretical traffic performance analysis of

an integrated voice-data virtual circuit packet switch, IEEE Trans. on Comm.,

COM-27, pp. 1104-1111.

AT&T (1961), Switching Systems, (American Telephone and Telegraph Co., New York,

NY).

Baran, P. (1964), On distributed communication networks, IEEE Trans. on Comm.

Systems, CS-12, pp. 1-9.

Bear, D. (1976), Princi les of Telecommunication - Traffic En ineerin , (P.

Peregrinus, Ltd., Stevenage, England.

Bene¥, V. E. (1965), Mathematical Theor of Connectin Networks and Te1e hone

,Traffic, (Academic Press, New York, NY .

Blackman, J. A. (1979), Switched communications for the Department of Defense,

IEEE Trans. on Comm., ,COM-27, pp. 1131-1137. -

Brockmeyer, E., H. L. Halstrom, and A. Jensen (1948), The life and works of A. K.

Erlang, Danish Acad. of Techn. Sci., Publ. No. 2, (Copenhagen~ Denmark).

Cooper, R. B. (1972), Introduction to Queueing Theory, (MacMillan Co., New York,

NY).

Fischer, M. J. (1979), Data performance in a system where-data packets are transmitted during voice silent periods - single channel case, IEEE Trans. on

Comm., COM-27, pp. 1371-1375.

Fischer, M. J., and T. C. Harris {1976), A model for evaluating the performance

of an integrated circuit - and packet-switched multiplex structure, IEEE

Trans. on Comm., COM-24, pp. 194-202.

Fry, T~, C. (1928), Probability and Its Engineering Uses, (D. Van Nostrand co.,

New York, NY).

Gallagher, E. F. (1979), Digital time division switching for military communications,

IEEE Trans on Comm., COM-27, pp. 1138-1143.

General Electric (1962), Tables of the Individual and Cumulative Terms of Poisson

pistribution, (D. Van Nostrand Co., New York, NY).

Haugen, R. B. (1979), Link systems with delay, IEEE Trans. on Comm., ,COM-27,

pp , 1261-1270.

Jacobaeus, C. (1950), A study on congestion in link systems, Ericsson Review, ~48, pp. 1-68.

68
Kleinrock, L. (1975)~ Queueing Systems, Volume I : Theory, (John Wiley and Sons,

New York,NY).

Kleinrock, L. (1976), Queueing Systems, Volume II : Computer Applications, (John

Wiley and Sons, New York, NY) .

Krevsky, S., L. H. Wagner, and A. P. Brogle (1972), The global digitally switched

communications system evolution, Proc. of Symp. on Computer-Communications

Networks and Teletraffic, ,Polytechnic Institute of Brooklyn, April, pp.

547-556.

Kuczura, A., and D. Bajaj (1977), Amethod of moments for the analysis of a switched

communication network1s performance, IEEE Trans. on Comm., COM-25, pp. 185-

193.

Lee, C. Y. (1955), Analysis of switching networks, Bell Syst. Tech. J., 34,

pp. 1287 -1315.

Lee, L., and J. A. Brzozowski (1966), An approximate method for computing blocking

probability in switching networks, IEEE Trans. on Comma Technology, COM-14,

pp. 85-93.

Levine, R. H. (1976), The evolving DCS: Introduction and overview, ICC Conf.

Record, Vol. II, Philadelphia, PA, June, pp. 33-2 to 33-4.

Lin, P. M., B. J. Leon, and C. R. Stewart (1978), Anaiysis of circuit-switched

networks employing operating - office control and spill-forward, IEEE Trans.

of Comm., COM-26, pp. 754-765.

Linfield, R. F. (1979), Control signaling in a military switching environment,

NTIA Report 79-13, U. S. Dept. of Commerce, National Telecommunications

and Information Administration, Institute for Telecommunication Sciences.

Available from NTIS, Springfield, VA 22161, Accession No. PB 292-377/AS.

Linfield, R. F., and M. Nesenbergs (1978), Access area signaling and switching:

concepts, issues and alternatives, NTIA Report 78-2, U. S. Dept. of Commerce,

National Telecommunications and Information Administration, Institute for

Telecommunication Sciences. Available from NTIS, Springfield, VA 22161,

Accession No. PB 283-464/AS~

Linfield, R. F., and M. Nesenbergs (1979), Switch element capacities in access

area digital switching systems, NTIA Report 79-26, U. S. Dept. of Commerce,

National Telecommunications and Information Administration, Institute for

Telecommunication Sciences. Available from NTIS, Springfield, VA 22161,

Accession No. PB

Lotze, A. (1963), 3rd Rep. Studies in Congestion Theory (CIRB), Institute for

Switching and Data Technics (Technical Univ. of Stuttgart, Stuttgart, Germany).

Manfield, D. R., and T. Downs (1979), On the one-moment analysis of telephone

traffic networks, IEEE Trans. on Comm., COM-27, pp. 1169-1174.

Molina, E. C. (1927), Application of the theory of probability to telephone

trunking problems, Bell Syst. Tech. J., §.., pp. 46'1-494.

69
National Bureau of Standards (1949), Tables of the Binomial Probabilit Distribution,

(U. S. Govt. Printing Office, Washington, DC , Applied Math~ Series, No.6.

Nesenbergs, M. (1979), A hybrid of Erlang Band C formulas and its applications,

IEEE Trans. on Comm.,COM-27, pp. 59-68.

Nesenbergs, M., and R. F. Linfield (1976), Parametric cost alternatives for local

digital distribution systems, OT Report 76-95, U. S. Dept. of Commerce,

Office of Telecommunications (nowNTIA), Institute for Telecommunication

Sciences. Available from NTIS, Springfield, VA 22161, Accession No. PB 258-

510/AS.

Riordan, J. (1962), Stochastic Service Systems, (John Wiley and Sons, New York,

NY) .

R5nnblom, N. (1959), Traffic loss of a circuit· group consisting of both-way

circuits, which is accessible for the internal and external traffic of a

subscriber group, Teleteknik (English Edition), No.2, pp. 79-92.

Saaty, T. L. (1961), Elements of Queueing Theory, (McGraw-Hill, New York, NY).

Schneider, K. A. (1979), Integrating voice and data on circuit - switched networks,

IEEE Trans. Aerospace and Electronic Systems, AES-15, pp. 481-493.

Schutzer, D. (1979), An analysis of a distributed switching network with integrated

voice and data in support of command and control, IEEE Trans. on Comm.,

COM-27, pp. 1124-1130.

Sherman, D. Ne (1971), Storage and delay estimates for asynchronous multiplexing

of data and speech, IEEE Trans. on Comma Technology, COM-19, pp. 551-555.

Siemens (1974), Telephone Traffic Theory Tables and Charts, (Siemens AG, MUnchen,

Germany), Second Edition,Part 1.

Tsuda, T.. , S. Hattori., R. Yatsuboshi, and K. Yamauchi (1979), An approach to

multi-service subscriber loop system using packetized voice/data terminals,

IEEE Trans. on Comm., COM-27, pp. 1112-1117.

Vena, P. A.. , and G. J. Coviello (1975), Integration of circuit/packet switching

bySENET (Slotted Envelope Network) concept, Conference Record of the

Natll. Telecom. Conf.(NTC 175), New Orleans, LA, pp.42/l3-42/17.

Weinstein, C. J., M. L.Malpass, andM. J. fischer (1978), Data traffic performance

of an integrated circuit - and packet-switched multiplex structure, Lincoln

Lab. Tech. Note 1978-41 (Lincoln Laboratory,Lexington, MA).

Wolman, E. (1965), On definitions of congestion in communication networks, Bell

Syst. Tech. J., 44, pp. 2271-2294.

70
FORM OT.29

(3-73)

u.s. DEPJ-.RTMENT OF COMMERCE OFFICE OF TELECOMMUNICATIONS

BIBLIOGRAPHIC DATA SHEET

1. PUBLICATION OR REPORT NO 2. Gov't Accession No.3. Recipienf's Accession No.

NTIA Report 80-34

~. TITLE AND SUBTITLE

Three Phases of Teletraffic Congestion in Military

Access Areas

7. AUTHOR(S)

M. Nesenbergs and R. F. Linfield

8. PERFORMING ORGANIZATION NAME AND ADDRESS

U. S. Department of Commerce

NTIA/ITS-4

325 Broadway

Boulder, Colorado 80303

11. Sponsorin. Or.anization Name and Address

U. S. Army Communications Systems Agency

Fort Monmouth, New Jersey 07703

1~. SUPPLEMENTARY NOTES

5. Pub lic at ion Date

February 1980

6. Perforrnin g Organization Code

NTIA/ITS-4

9. Project/1'ask/Work Unit No.

10. Contract /Grant No.

12. Type of keport and Period Covered

13.

15. ABSTRACT (A 200-word or less factual summary of most significant information. If document' includes a signtticent

bibliography of literature survey, mention it here.')

In military base access areas, akin to many commercial installations, the

telecommunications traffic passes through several concentration and switching

stages. Loads to server facilities are formed through mergers and branch outs

of off~red traffic substreams. Blocking of calls is known to occur in many

ways throughout the existing access area networks.

In this report, an effort is made to represent the access area grades of service (i.e., the probabilities of blocking for different substream~) in more

realistic ways than before. The message flow process is structured into three

representative contention phases. The three phases are realistic and occur

often in military networks. All three phases apparently possess queueing models

and analytical properties distinct from the conventional Engset, Erlang, and

other classical models. Their blocking probabilities also differ significantly.

One of the three models appears tractable only through bounds and asympto- tically~ tight approximations. The otber two models are shown to permit formal

solutions. Given an access area network, the three blocking probabilities may

be applied individually or in a variety of combinations. The paper demonstrates several applications to access area telephony.

17. AVAILABILITY STATEMENT

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18. Securi ty Class (Thie report)

Unclassified

19. Security Class (Thi« pa~)

·Unclassified

20. Number of page s

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