# On the Discrete-TimeG

Authors: Jianjun Li; Liwei Liu
Journal: Discrete Dynamics in Nature and Society
 (2016)

---

## Abstract

A discrete-timeG
e
o
/
G
/
1 queue with vacations in random environment is analyzed. Using the method of supplementary variable, we give the probability generating function (PGF) of the stationary queue length distribution at arbitrary epoch. The PGF of the stationary sojourn time distribution is also derived. And we present the various performance measures such as mean number of customers in the system, mean length of the type-i cycle, and mean time that the system resides in phase 0. In addition, we show that the M
/
G
/
1 queue with vacations in random environment can be approximated by its discrete-time counterpart. Finally, we present some special cases of the model and numerical examples.


---

## Body

## 1. Introduction

During the last three decades queueing systems with vacations have been largely studied; see, for example, the surveys (Doshi [1]) and the monographs (Takagi [2] and Tian and Zhang [3]). Queueing systems with vacations are characterized by the feature that each time a busy period ends and the system becomes empty, the server starts a vacation of random length of time. These queueing models have been applied in many fields such as production systems and computer and communication systems. Doshi [1] gives a large number of examples. See also Takagi [2].Due to the applicability in the field of computer and communication systems in which time is slotted, discrete-time queueing models have been widely used over the past several years. For a detailed discussion and applications of discrete-time queues, see, for example, Hunter [4], Bruneel and Kim [5], and Woodward [6]. The first work on discrete-time queues is due to Meisling [7]. Since then, parallel to the continuous-time vacation queues, discrete-time queues have been intensively studied by many researchers. The early results of discrete-time queues can be found in the book of Hunter [4]. Subsequently, Takagi [8] provided a detailed analysis on G
e
o
/
G
/
1 queues with a variety of vacation policies. The work about a G
e
o
/
G
/
1 queue with multiple adaptive vacations was considered by Zhang and Tian [9]. Alfa [10] discussed a model with nonexhaustive service in which both vacation time and service time follow phase type distributions. Recently, Atencia and Moreno [11] studied a discrete-time G
e
o
/
G
/
1 retrial queue, where the server is subject to starting failures. Li and Tian [12] considered a G
I
/
G
e
o
/
1 queue with working vacations and vacation interruption. Using the matrix-analytic method, Tao et al. [13] analyzed the G
I
/
G
e
o
/
1 queue with Bernoulli-schedule-controlled vacation and vacation interruption. Vijaya Laxmi et al. [14] analyzed a discrete-time working vacation queue with balking.In most of the literature on queueing theory, both arrival rate and service rate are homogeneous. But, in real life, the parameters in the queueing models may not be constant; they may change with changes of the environment. Yechiali and Naor [15] study a two-level modification of the M
/
M
/
1 queueing model, where the rate of arrival and the service capacity are subject to Poisson alternations. Their work is considered to be the first systematic work on queueing system in random environment. The M
/
G
/
1 queue in a two-phase random environment was studied by Neuts [16], Boxma and Kurkova [17], Huang and Lee [18], and so on. Recently, Cordeiro and Kharoufeh [19] discussed an unreliable M
/
M
/
1 retrial queue whose arrival, service, failure, repair, and retrial rates are all modulated by an exogenous random environment. B. Kim and J. Kim [20] considered a single server queue with Markov modulated service rates and impatient customers. An M
/
M
/
1 queue in random environment with disasters was investigated by Paz and Yechiali [21] and Jiang et al. [22] extended their work to M
/
G
/
1 queue.To the best of our knowledge, there has been no research on the discrete-time queueing system with vacations in random environment. However, in many practical applications like production, these queueing systems may be utilized. For example, in production system, if there is no workload to be processed, the server has a vacation period. When it returns from vacation, if there are one or more customers in the system, it continues to serve customers until the system becomes empty. But the new service rate may change with the changes of the customers’ arrival rate, environmental conditions, and operator experience. This motivated us to study theG
e
o
/
G
/
1 queue with vacations in random environment in this paper.The rest of the paper is organized as follows. Section2 is devoted to the model description. Section 3 obtains the probability generating function of the steady-state queue size distribution at an arbitrary epoch. Section 4 presents the various performance measures such as mean number of customers in the system, mean length of the type-i cycle, and PGF of the steady-state sojourn time distribution. Section 5 gives the relationship between the discrete-time system and its continuous-time counterpart. Some special cases are presented in Section 6. Numerical results are presented in Section 7 followed by conclusions in Section 8.


## 2. Model Description

We consider a discrete-time vacation queue operating in random environment where the time axis is segmented into slots of equal length. It is assumed that the time axis is marked by0,1
,
…
,
t
,
… and all queueing activities occur only at the slot boundaries. We consider late arrival system (LAS) policy in our queueing system. That is, we assume that the arrivals occur in (
t
-
,
t
), the departures occur in (
t
,
t
+
), and the vacation completion occurs just at the instant t.When the system operates in phasei, i
=
1,2
,
…
,
n, customers arrive according to geometrical process with rate λ
i, where λ
i is the probability that a customer arrives in a slot, and the service times are independent and identically distributed according to a general distribution {
s
i
,
k
}
k
=
1
∞ with probability generating function S
i
(
z
)
=
∑
k
=
1
∞
s
i
,
k
z
k and mean 1
/
μ
i. Customers are served according to the first-come, first-served (FCFS) discipline. Each time a busy period ends and the system becomes empty, the server leaves for a vacation of random length V, which is geometrically distributed with parameter θ, causing the system to move to vacation phase 0, where θ is the probability that vacation ends in a slot. In vacation phase 0, the arrivals occur according to a geometrical process of rate λ
0. When the server returns, if it finds no customer waiting, it goes on another vacation. Otherwise, the system moves from the vacation phase 0 to some operative phase i with probability q
i, i
=
1,2
,
…
,
n, where q
i
≥
0 and ∑
i
=
1
n
q
i
=
1. The interarrival times, the individual service, and vacation durations are assumed to be mutually independent. Throughout the rest of the paper, we denote p
¯
=
1
-
p.


## 3. Steady-State Queue Size Distribution

In operative phasei, i
=
1,2
,
…
,
n, it is easily seen that the system acts as the classical G
e
o
/
G
/
1 queue with geometrical arrival rate λ
i and service rate μ
i. So as long as ρ
i
=
λ
i
/
μ
i
<
1, i
=
1,2
,
…
,
n, this system that we consider is stable. Assume that the condition for stability of the system is fulfilled. Next, we analyze the steady-state queue size distribution.At timet
+, the system can be described by the process {
(
L
(
t
+
)
,
J
(
t
+
)
,
S
+
(
t
+
)
)
,
t
=
0,1
,
2
,
…
}, where L
(
t
+
) denotes the number of customers in the system and J
(
t
+
) represents the phase in which the system operates. If L
(
t
+
)
≥
1 and J
(
t
+
)
=
i, i
=
1,2
,
…
,
n, S
+
(
t
+
)
=
S
i
+
(
t
+
) represents the remaining service time of the customer currently being served during phase i. Then {
(
L
(
t
+
)
,
J
(
t
+
)
,
S
+
(
t
+
)
)
,
t
=
0,1
,
2
,
…
} is a Markov process with the state space expressed as (1)
S
=
m
,
0
,
m
≥
0
∪
m
,
i
,
k
,
m
≥
1
,
k
≥
1
,
i
=
1,2
,
…
,
n
.Since we are interested in the stationary behavior of the system, define(2)Pm,0=limt→∞PLt+=m,Jt+=0,m≥0,Pm,ik=limt→∞PLt+=m,Jt+=i,Si+t+=k,m≥1,k≥1,i=1,2,…,n.The balance equations for the stationary distribution of the system are(3)P0,0=λ¯0P0,0+∑i=1nλ¯iP1,i1,(4)Pm,0=λ¯0θ¯Pm,0+λ0θ¯Pm-1,0,m≥1,(5)Pm,ik=λi1-δm,1Pm-1,ik+1+λ¯iPm,ik+1+λisi,kPm,i1+λ¯isi,kPm+1,i1+θqisi,kλ¯0Pm,0+λ0Pm-1,0,k≥1,m≥1,i=1,2,…,n,where δ
m
,
1 is Kronecker’s delta.To solve the above equations (3)–(5) let us define the following PGFs: (6)P0z=∑m=0∞Pm,0zm,Pik,z=∑m=1∞Pm,ikzm,k≥1,i=1,2,…,n,Pi∗x,z=∑k=1∞Pik,zxk=∑k=1∞∑m=1∞Pm,ikzmxk.From (4), we obtain(7)Pm,0=λ0θ¯1-λ¯0θ¯Pm-1,0=λ0θ¯1-λ¯0θ¯mP0,0,and then, we have(8)P0z=∑m=0∞λ0θ¯1-λ¯0θ¯mzmP0,0=1-λ¯0θ¯1-λ¯0θ¯-λ0θ¯zP0,0.Multiplying both sizes of (5) by z
m and then taking summation over all possible values of m
≥
1, we get(9)Pik,z=λiz+λ¯iPik+1,z+λiz+λ¯iz-1si,kPi1,z+θqisi,kλ¯0+λ0zP0z-λ¯isi,kP1,i1-θqisi,kλ¯0P0,0.Again multiplying both sides of (9) by x
k and then taking summation over all possible values of k
≥
1, we get(10)1-αizx-1Pi∗x,z=z-1αizSix-zPi1,z-λ¯iSixP1,i1-θqiλ¯0SixP0,0+θqiλ¯0+λ0zSixP0z,where α
i
(
z
)
=
λ
i
z
+
λ
¯
i.Settingx
=
α
i
(
z
) in (10), we get(11)αizSiαiz-zPi1,z=zSiαiz·λ¯iP1,i1+θqiλ¯0P0,0-θqiλ¯0+λ0zP0z.Setting z
=
1 in (11), we get(12)P1,i1=θqiP01-λ¯0P0,0λ¯i.Substituting (12) into (11), we have(13)Pi1,z=zSiαizθqiP01-θqiλ¯0+λ0zP0zαizSiαiz-z.Substituting (12) and (13) into (10), we obtain(14)Pi∗x,z=xzθqiP01-λ¯0+λ0zP0zSix-Siαizx-αizSiαiz-z.Setting x
=
1 in (14), we get(15)Pi∗1,z=zθqiP01-λ¯0+λ0zP0z1-Siαiz1-αizSiαiz-z.Then from (8) and (15), we get the distribution of the number of customers in the system having PGF(16)
P
z
=
P
0
z
+
∑
i
=
1
n
P
i
∗
1
,
z
=
P
0
z
+
∑
i
=
1
n
z
θ
q
i
P
0
1
-
λ
¯
0
+
λ
0
z
P
0
z
1
-
S
i
α
i
z
1
-
α
i
z
S
i
α
i
z
-
z
=
1
-
λ
¯
0
θ
¯
1
-
λ
¯
0
θ
¯
-
λ
0
θ
¯
z
+
1
-
λ
¯
0
θ
¯
λ
0
1
-
z
z
1
-
λ
¯
0
θ
¯
-
λ
0
θ
¯
z
·
∑
i
=
1
n
q
i
1
-
S
i
α
i
z
1
-
α
i
z
S
i
α
i
z
-
z
P
0,0
.PuttingD
(
z
)
=
[
1
-
α
i
z
]
[
S
i
(
α
i
(
z
)
)
-
z
], N
(
z
)
=
z
[
P
0
1
-
λ
¯
0
+
λ
0
z
P
0
z
]
[
1
-
S
i
(
α
i
(
z
)
)
]. Let z
→
1 in (15); we obtain by L’Hospital’s Rule(17)Pi∗1,1=limz→1Pi∗1,z=θqiN′′1D′′1,where(18)D′′1=2λiμi-λiμi,N′′1=2λiλ01-λ¯0θ¯μiθ2P0,0.Substituting (18) into (17), we get(19)Pi∗1,1=qiλ01-λ¯0θ¯θμi-λiP0,0.From normalizing condition that can be written as(20)P01+∑i=1nPi∗1,1=1,we get on simplification(21)P0,0=1λ0β,where(22)β=1-λ¯0θ¯λ0θ+∑i=1nqi1-λ¯0θ¯θμi-λi.Now we summarize the results of this section in the following theorem.Theorem 1.
If the stationary conditionρ
i
=
λ
i
/
μ
i
<
1, i
=
1,2
,
…
,
n, holds, the PGF of the distribution of the number of customers in the system is given by(23)
P
z
=
1
-
λ
¯
0
θ
¯
1
-
λ
¯
0
θ
¯
-
λ
0
θ
¯
z
+
1
-
λ
¯
0
θ
¯
λ
0
1
-
z
z
1
-
λ
¯
0
θ
¯
-
λ
0
θ
¯
z
·
∑
i
=
1
n
q
i
1
-
S
i
α
i
z
1
-
α
i
z
S
i
α
i
z
-
z
1
λ
0
β
,where β is given by (22).


## 4. Performance Measures

In this section, some important performance measures of the system will be provided.


### 4.1. Mean Number of Customers in the System

The mean number of customers in the system(
L
) is obtained as(24)L=dPzdzz=1=dP0zdzz=1+∑i=1ndPi∗1,zdzz=1=1-λ¯0θ¯θ¯θ2β+1-λ¯0θ¯β·∑i=1nqiN1′′′1D1′′1-N1′′1D1′′′13D1′′12,where D
1
(
z
)
=
(
1
-
λ
¯
0
θ
¯
-
λ
0
θ
¯
z
)
[
1
-
α
i
z
]
[
S
i
(
α
i
(
z
)
)
-
z
], N
1
(
z
)
=
z
(
1
-
z
)
[
1
-
S
i
(
α
i
(
z
)
)
].After some calculations we get(25)D1′′1=2θλi1-λiμi,D1′′′1=6λ0θ¯λiλiμi-1-3θλi3βi2-1μi,N1′′1=2λiμi,N1′′′1=6λiμi+3λi2βi2-1μi,where β
i
(
2
) is the 2nd moment of the service time. Substituting the above results into (24), we have(26)L=1-λ¯0θ¯θ¯θ2β+1-λ¯0θ¯θ2β·∑i=1nqi2μi-λiθ+λ0θ¯+θλiμi2βi2-μi2μi-λi2.


### 4.2. Mean Time That the System Resides in Phase 0

We denoteA as the interarrival time during a vacation period, which has geometrical distribution with parameter λ
0. Then, the probability that there is no customer arrival during a vacation period is given by (27)PA>V=∑k=1∞PV=kPA>k∣V=k=λ¯0θ1-λ¯0θ¯;that is to say, the probability that at least one customer arrives during a vacation period is λ
0
/
(
1
-
λ
¯
0
θ
¯
). We can easily find that the number of vacations in phase 0 is geometrically distributed with parameter λ
0
/
(
1
-
λ
¯
0
θ
¯
) and the mean time of each vacation is 1
/
θ. Hence, the mean time that the system resides in phase 0, denoted by T
0, is given by(28)T0=∑k=1∞λ¯0θ1-λ¯0θ¯k-1·λ01-λ¯0θ¯·kθ=1-λ¯0θ¯λ0θ,which is just the product of the mean number of vacations in phase 0 and the mean time of each vacation.


### 4.3. Mean Number of Customers in the System at the End of Phase 0

Leta
m be the probability that m customers arrive during a vacation V, and let b
m represent the probability that m customers arrive during phase 0. Then (29)am=∑k=m∞θ¯k-1θCkmλ0mλ¯0k-m,bm=∑k=1∞λ¯0θ1-λ¯0θ¯k-1am=1-λ¯0θ¯λ0am.Then, the mean number of customers in the system at the end of phase 0, denoted by N
0, is given by(30)N0=∑m=1∞m·bm=1-λ¯0θ¯λ0·∑m=1∞m·am=1-λ¯0θ¯θ,which is the product of the mean time that the system resides in phase 0, that is, T
0, and the mean number of customers that arrive to the system per unit time during phase 0, that is, λ
0.


### 4.4. Mean Time That the System Resides in Phasei

We denoteT
i as the mean time that the system resides in phase i. Then(31)Ti=∑m=1∞bmmμi-λi=1-λ¯0θ¯θμi-λi,where 1
/
(
μ
i
-
λ
i
) is the mean busy period that one customer induces in phase i. Actually, the mean time that the system resides in phase i is the product of the mean number of customers in the system at the end of phase 0 and the mean busy period that one customer induces in phase i.


### 4.5. Probability That the System Resides in Phase 0

From (8) and (21), the probability that the system resides in phase 0 is given by(32)P01=1-λ¯0θ¯λ0θβ.Intuitively, in the time interval between two consecutive epochs at which phase 0 begins, the probability that the system resides in phase 0 is equal to the proportion of time the system resides in phase 0. Then, we have(33)P01=1-λ¯θ¯/λ0θ1-λ¯θ¯/λ0θ+∑i=1nqi1-λ¯θ¯/θμi-λi=1-λ¯θ¯λ0θβ,where 1
-
λ
¯
θ
¯
/
λ
0
θ is the mean time that the system resides in vacation phase and ∑
i
=
1
n
q
i
(
1
-
λ
¯
θ
¯
)
/
θ
(
μ
i
-
λ
i
) is the mean time that the system resides in operative phase.


### 4.6. Probability That the System Resides in Phasei

From (19) and (21), the probability that the system resides in phase i is given by(34)Pi∗1,1=qi1-λ¯θ¯θμi-λiβ,which can be intuitively explained in a way similar to the one described above for the probability that the system resides in phase 0.


### 4.7. Mean Length of the Type-i Cycle

The length of type-i cycle, denoted by C
i, is the length of time from the beginning of the last phase i to the beginning of the next phase i, i
=
0,1
,
…
,
n. Then, there are (
n
+
1
) types of cycles. From (28) and (31), we can easily get the mean length of the type-0 cycle, denoted by E
(
C
0
), as equal to β. Now, we calculate the mean length of the type-i cycle, denoted by E
(
C
i
), i
=
1,2
,
…
,
n.We can easily find that the times of the system visiting phase0 during type-i cycle are distributed with geometric distribution with parameter q
i, and the mean length of type-i cycle during which the system visits phase 0 k times is (35)1-λ¯0θ¯λ0θ+1-λ¯0θ¯θμi-λi+k-1·1-λ¯0θ¯λ0θ+∑j=1,j≠inqj1-λ¯0θ¯θμj-λj1-qi.Then, the mean length of the type-i cycle is given by(36)
E
C
i
=
∑
k
=
1
∞
1
-
q
i
k
-
1
q
i
1
-
λ
¯
0
θ
¯
λ
0
θ
+
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
k
-
1
·
1
-
λ
¯
0
θ
¯
λ
0
θ
+
∑
j
=
1
,
j
≠
i
n
q
j
1
-
λ
¯
0
θ
¯
θ
μ
j
-
λ
j
1
-
q
i
=
1
-
λ
¯
0
θ
¯
λ
0
θ
+
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
∑
k
=
1
∞
k
-
1
1
-
q
i
k
-
1
·
q
i
1
-
λ
¯
0
θ
¯
λ
0
θ
+
β
-
1
-
λ
¯
0
θ
¯
λ
0
θ
-
q
i
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
·
1
1
-
q
i
=
1
-
λ
¯
0
θ
¯
λ
0
θ
+
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
1
-
q
i
q
i
1
-
λ
¯
0
θ
¯
λ
0
θ
-
q
i
1
-
q
i
+
β
-
q
i
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
·
1
1
-
q
i
=
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
1
q
i
β
-
q
i
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
=
β
q
i
.


### 4.8. PGF of the Stationary Sojourn Time Distribution of an Arbitrary Customer

The stationary sojourn times of an arbitrary customer, a customer who arrives in the state(
m
,
0
) and a customer who arrives in the state (
m
,
i
,
k
), i
=
1,2
,
…
,
n, respectively, are denoted by W, W
m
,
0, and W
m
,
i
,
k, with their respective PGFs W
(
z
), W
m
,
0
(
z
), and W
m
,
i
,
k
(
z
). We also define W
m
,
i and W
m
,
i
(
z
), respectively, as the sum of the (
m
+
1
) customers’ service times and its PGF, where i
=
1,2
,
…
,
n.When a customer arrives in state(
m
,
0
), m
=
0,1
,
2
,
…, his sojourn time until departure is the sum of the remaining vacation time and the (
m
+
1
) customers’ service times. Because the vacation time is distributed with a geometrical distribution, the remaining vacation time is identically distributed with vacation time. Then, we have(37)Wm,0z=∑i=1nqiVzWm,iz=∑i=1nqiSizm+1θz1-θ¯z,where V
(
z
) is the PGF of the vacation time distribution and V
(
z
)
=
θ
z
/
(
1
-
θ
¯
z
).When a customer arrives in state(
m
,
i
,
k
), m
≥
1, k
≥
1, i
=
1,2
,
…, the PGF of the customer’s stationary sojourn time until departure is given by(38)Wm,i,kz=Ezk+Wm-1,i=zkSizm.Hence, we have the PGF of the stationary sojourn time distribution of an arbitrary customer as follows:(39)Wz=∑m=0∞Pm,0Wm,0z+∑i=1n∑m=1∞∑k=1∞Pm,ikWm,i,kz=∑m=0∞λ0θ¯1-λ¯0θ¯mP0,0∑i=1nqiSizm+1θz1-θ¯z+∑i=1n∑m=1∞∑k=1∞Pm,ikzkSizm=θzP0,01-θ¯z∑i=1nqi∑m=0∞λ0θ¯1-λ¯0θ¯mSizm+1+∑i=1nPi∗z,Siz=∑i=1nqiSiz1-λ¯0θ¯-λ0θ¯Siz1-λ¯0θ¯θz1-θ¯zλ0β+∑i=1nPi∗z,Siz.


## 4.1. Mean Number of Customers in the System

The mean number of customers in the system(
L
) is obtained as(24)L=dPzdzz=1=dP0zdzz=1+∑i=1ndPi∗1,zdzz=1=1-λ¯0θ¯θ¯θ2β+1-λ¯0θ¯β·∑i=1nqiN1′′′1D1′′1-N1′′1D1′′′13D1′′12,where D
1
(
z
)
=
(
1
-
λ
¯
0
θ
¯
-
λ
0
θ
¯
z
)
[
1
-
α
i
z
]
[
S
i
(
α
i
(
z
)
)
-
z
], N
1
(
z
)
=
z
(
1
-
z
)
[
1
-
S
i
(
α
i
(
z
)
)
].After some calculations we get(25)D1′′1=2θλi1-λiμi,D1′′′1=6λ0θ¯λiλiμi-1-3θλi3βi2-1μi,N1′′1=2λiμi,N1′′′1=6λiμi+3λi2βi2-1μi,where β
i
(
2
) is the 2nd moment of the service time. Substituting the above results into (24), we have(26)L=1-λ¯0θ¯θ¯θ2β+1-λ¯0θ¯θ2β·∑i=1nqi2μi-λiθ+λ0θ¯+θλiμi2βi2-μi2μi-λi2.


## 4.2. Mean Time That the System Resides in Phase 0

We denoteA as the interarrival time during a vacation period, which has geometrical distribution with parameter λ
0. Then, the probability that there is no customer arrival during a vacation period is given by (27)PA>V=∑k=1∞PV=kPA>k∣V=k=λ¯0θ1-λ¯0θ¯;that is to say, the probability that at least one customer arrives during a vacation period is λ
0
/
(
1
-
λ
¯
0
θ
¯
). We can easily find that the number of vacations in phase 0 is geometrically distributed with parameter λ
0
/
(
1
-
λ
¯
0
θ
¯
) and the mean time of each vacation is 1
/
θ. Hence, the mean time that the system resides in phase 0, denoted by T
0, is given by(28)T0=∑k=1∞λ¯0θ1-λ¯0θ¯k-1·λ01-λ¯0θ¯·kθ=1-λ¯0θ¯λ0θ,which is just the product of the mean number of vacations in phase 0 and the mean time of each vacation.


## 4.3. Mean Number of Customers in the System at the End of Phase 0

Leta
m be the probability that m customers arrive during a vacation V, and let b
m represent the probability that m customers arrive during phase 0. Then (29)am=∑k=m∞θ¯k-1θCkmλ0mλ¯0k-m,bm=∑k=1∞λ¯0θ1-λ¯0θ¯k-1am=1-λ¯0θ¯λ0am.Then, the mean number of customers in the system at the end of phase 0, denoted by N
0, is given by(30)N0=∑m=1∞m·bm=1-λ¯0θ¯λ0·∑m=1∞m·am=1-λ¯0θ¯θ,which is the product of the mean time that the system resides in phase 0, that is, T
0, and the mean number of customers that arrive to the system per unit time during phase 0, that is, λ
0.


## 4.4. Mean Time That the System Resides in Phasei

We denoteT
i as the mean time that the system resides in phase i. Then(31)Ti=∑m=1∞bmmμi-λi=1-λ¯0θ¯θμi-λi,where 1
/
(
μ
i
-
λ
i
) is the mean busy period that one customer induces in phase i. Actually, the mean time that the system resides in phase i is the product of the mean number of customers in the system at the end of phase 0 and the mean busy period that one customer induces in phase i.


## 4.5. Probability That the System Resides in Phase 0

From (8) and (21), the probability that the system resides in phase 0 is given by(32)P01=1-λ¯0θ¯λ0θβ.Intuitively, in the time interval between two consecutive epochs at which phase 0 begins, the probability that the system resides in phase 0 is equal to the proportion of time the system resides in phase 0. Then, we have(33)P01=1-λ¯θ¯/λ0θ1-λ¯θ¯/λ0θ+∑i=1nqi1-λ¯θ¯/θμi-λi=1-λ¯θ¯λ0θβ,where 1
-
λ
¯
θ
¯
/
λ
0
θ is the mean time that the system resides in vacation phase and ∑
i
=
1
n
q
i
(
1
-
λ
¯
θ
¯
)
/
θ
(
μ
i
-
λ
i
) is the mean time that the system resides in operative phase.


## 4.6. Probability That the System Resides in Phasei

From (19) and (21), the probability that the system resides in phase i is given by(34)Pi∗1,1=qi1-λ¯θ¯θμi-λiβ,which can be intuitively explained in a way similar to the one described above for the probability that the system resides in phase 0.


## 4.7. Mean Length of the Type-i Cycle

The length of type-i cycle, denoted by C
i, is the length of time from the beginning of the last phase i to the beginning of the next phase i, i
=
0,1
,
…
,
n. Then, there are (
n
+
1
) types of cycles. From (28) and (31), we can easily get the mean length of the type-0 cycle, denoted by E
(
C
0
), as equal to β. Now, we calculate the mean length of the type-i cycle, denoted by E
(
C
i
), i
=
1,2
,
…
,
n.We can easily find that the times of the system visiting phase0 during type-i cycle are distributed with geometric distribution with parameter q
i, and the mean length of type-i cycle during which the system visits phase 0 k times is (35)1-λ¯0θ¯λ0θ+1-λ¯0θ¯θμi-λi+k-1·1-λ¯0θ¯λ0θ+∑j=1,j≠inqj1-λ¯0θ¯θμj-λj1-qi.Then, the mean length of the type-i cycle is given by(36)
E
C
i
=
∑
k
=
1
∞
1
-
q
i
k
-
1
q
i
1
-
λ
¯
0
θ
¯
λ
0
θ
+
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
k
-
1
·
1
-
λ
¯
0
θ
¯
λ
0
θ
+
∑
j
=
1
,
j
≠
i
n
q
j
1
-
λ
¯
0
θ
¯
θ
μ
j
-
λ
j
1
-
q
i
=
1
-
λ
¯
0
θ
¯
λ
0
θ
+
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
∑
k
=
1
∞
k
-
1
1
-
q
i
k
-
1
·
q
i
1
-
λ
¯
0
θ
¯
λ
0
θ
+
β
-
1
-
λ
¯
0
θ
¯
λ
0
θ
-
q
i
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
·
1
1
-
q
i
=
1
-
λ
¯
0
θ
¯
λ
0
θ
+
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
1
-
q
i
q
i
1
-
λ
¯
0
θ
¯
λ
0
θ
-
q
i
1
-
q
i
+
β
-
q
i
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
·
1
1
-
q
i
=
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
+
1
q
i
β
-
q
i
1
-
λ
¯
0
θ
¯
θ
μ
i
-
λ
i
=
β
q
i
.


## 4.8. PGF of the Stationary Sojourn Time Distribution of an Arbitrary Customer

The stationary sojourn times of an arbitrary customer, a customer who arrives in the state(
m
,
0
) and a customer who arrives in the state (
m
,
i
,
k
), i
=
1,2
,
…
,
n, respectively, are denoted by W, W
m
,
0, and W
m
,
i
,
k, with their respective PGFs W
(
z
), W
m
,
0
(
z
), and W
m
,
i
,
k
(
z
). We also define W
m
,
i and W
m
,
i
(
z
), respectively, as the sum of the (
m
+
1
) customers’ service times and its PGF, where i
=
1,2
,
…
,
n.When a customer arrives in state(
m
,
0
), m
=
0,1
,
2
,
…, his sojourn time until departure is the sum of the remaining vacation time and the (
m
+
1
) customers’ service times. Because the vacation time is distributed with a geometrical distribution, the remaining vacation time is identically distributed with vacation time. Then, we have(37)Wm,0z=∑i=1nqiVzWm,iz=∑i=1nqiSizm+1θz1-θ¯z,where V
(
z
) is the PGF of the vacation time distribution and V
(
z
)
=
θ
z
/
(
1
-
θ
¯
z
).When a customer arrives in state(
m
,
i
,
k
), m
≥
1, k
≥
1, i
=
1,2
,
…, the PGF of the customer’s stationary sojourn time until departure is given by(38)Wm,i,kz=Ezk+Wm-1,i=zkSizm.Hence, we have the PGF of the stationary sojourn time distribution of an arbitrary customer as follows:(39)Wz=∑m=0∞Pm,0Wm,0z+∑i=1n∑m=1∞∑k=1∞Pm,ikWm,i,kz=∑m=0∞λ0θ¯1-λ¯0θ¯mP0,0∑i=1nqiSizm+1θz1-θ¯z+∑i=1n∑m=1∞∑k=1∞Pm,ikzkSizm=θzP0,01-θ¯z∑i=1nqi∑m=0∞λ0θ¯1-λ¯0θ¯mSizm+1+∑i=1nPi∗z,Siz=∑i=1nqiSiz1-λ¯0θ¯-λ0θ¯Siz1-λ¯0θ¯θz1-θ¯zλ0β+∑i=1nPi∗z,Siz.


## 5. Relation to the Continuous-Time System

In this section, we analyze the relationship between the discrete-time system and its continuous-time counterpart. The continuous-time system can be approximated by the discrete-time system. If we assume that time is slotted into intervals of constant lengthΔ, when Δ goes to zero, the approximation tends to the exact value.We consider the continuous-timeM
/
G
/
1 queue with vacations in multiphase random environment. In phase i, i
=
1,2
,
…
,
n, customers arrive according to a Poisson stream with rate λ
~
i and service times are independent and identically distributed with general distribution function B
i
(
x
), LST B
i
∗
(
s
), and finite means 1
/
μ
~
i. In phase 0, customers arrive according to a Poisson stream with rate λ
~
0. Vacation times are independent exponential random variables with parameter θ
~. If we assume time is slotted into intervals of constant length Δ, the continuous-time system can be approximated by a discrete-time system for which (40)λ0=λ~0Δ,λi=λ~iΔ,μi=μ~iΔ,i=1,2,…,n,θ=θ~Δ,si,k=∫k-1ΔkΔdBix,i=1,2,…,n,k≥1,where Δ is sufficiently small so that λ
0, λ
i, μ
i, and θ are probabilities.Using the same technique with Yang and Li [23], it is not difficult to show that l
i
m
Δ
→
0
P
(
z
) is the probability generating function of the number of customers in the M
/
G
/
1 queue with vacations in multiphase random environment.From (8), we have(41)limΔ→0P0z=limΔ→01-λ¯0θ¯1-λ¯0θ¯-λ0θ¯zP0,0=limΔ→01-1-λ~0Δ1-θ~Δ1-1-λ~0Δ1-θ~Δ-λ~0Δ1-θ~ΔzP0,0=limΔ→0λ~0Δ+θ~Δ-λ~0θ~Δ2λ~0Δ+θ~Δ-λ~0θ~Δ2-λ~0zΔ+λ~0θ~zΔ2P0,0=limΔ→0λ~0+θ~-λ~0θ~Δλ~0+θ~-λ~0θ~Δ-λ~0z+λ~0θ~zΔP0,0.Since (42)limΔ→0Si′1Δ=limΔ→0∑k=1∞kΔsi,k=limΔ→0∑k=1∞kΔBikΔ-Bik-1Δ=∫0∞xdBix=1μ~i,limΔ→0P0,0=limΔ→01λ0β=limΔ→01λ01-λ¯0θ¯/λ0θ+∑i=1nqi1-λ¯0θ¯/θμi-λi=limΔ→01λ~0Δ1-1-λ~0Δ1-θ~Δ/λ~0Δθ~Δ+∑i=1nqi1-1-λ~0Δ1-θ~Δ/θ~Δ1/Si′1-λ~iΔ=limΔ→01λ~0+θ~-λ~0θ~Δ/θ~+∑i=1nqiλ~0λ~0+θ~-λ~0θ~Δ/θ~1/Si′1Δ-λ~i=limΔ→01λ~0+θ~-λ~0θ~Δ/θ~+∑i=1nqiλ~0λ~0+θ~-λ~0θ~Δ/θ~μ~i-λ~i=1λ~0+θ~/θ~+∑i=1nqiλ~0λ~0+θ~/θ~μ~i-λ~i=P~0,0,we obtain(43)limΔ→0P0z=λ~0+θ~λ~01-z+θ~P~0,0.Since (44)limΔ→0Siαiz=limΔ→0Siλ¯i+λiz=limΔ→0∑k=1∞1-λ~iΔ+λ~iΔzk·BikΔ-Bik-1Δ=limΔ→0∑k=1∞1-λ~i1-zΔ1/ΔkΔ·BikΔ-Bik-1Δ=∫0∞e-λ~i1-zxdBix=Bi∗λ~i1-z,we get(45)limΔ→0Pi∗1,z=limΔ→01-λ¯0θ¯λ0qiz1-z1-Siαiz1-αizSiαiz-z1-λ¯0θ¯-λ0θ¯zP0,0=limΔ→0λ~0Δ+θ~Δ-λ~0θ~Δ2λ~0Δqiz1-z1-Siαizλ~iΔ-λ~iΔzSiαiz-zλ~0Δ+θ~Δ-λ~0θ~Δ2-λ~0zΔ+λ~0θ~zΔ2P0,0=λ~0+θ~λ~0qiz1-Bi∗λ~i1-zλ~iλ~0+θ~-λ~0zBi∗λ~i1-z-zP~0,0=P~iz.From (43) and (45), we obtain the PGF of the number of customers in the M
/
G
/
1 queue with vacations in multiphase random environment as follows:(46)P~z=P~0z+∑i=1nP~iz=λ~0+θ~P~0,0λ~01-z+θ~+λ~0+θ~λ~0zP~0,0λ~01-z+θ~∑i=1nqi1-Bi∗λ~i1-zλ~iBi∗λ~i1-z-z.


## 6. Special Cases

### 6.1. TheG
e
o
/
G
/
1 Queue with Vacations

When the system is homogeneous, that is,λ
i
=
λ and μ
i
=
μ, the model translates into a regular G
e
o
/
G
/
1 queue with multiple vacations. From (23), P
(
z
) reduces to(47)Pz=θ1-ρ1-zSαz1-θ¯αzSαz-z,which is the PGF of the number of customers in the regular G
e
o
/
G
/
1 queue with multiple vacations, where S
(
z
) is the PGF of the service time distribution, α
(
z
)
=
λ
¯
+
λ
z. Expression (47) is in agreement with that in Tian and Zhang [3].


### 6.2. TheG
e
o
/
G
e
o
/
1 Queue with Vacations in Random Environment

Suppose that the service times in phasei, i
=
1,2
,
…
,
n, follow the geometric distribution with finite means 1
/
μ
i in the model; then S
i
(
z
)
=
μ
i
z
/
1
-
μ
¯
i
z in (23), and (26) yields(48)L=1-λ¯0θ¯θ¯θ2β+1-λ¯0θ¯θ2β∑i=1nqiμi-λiθ+λ0θ¯+θλiμ¯iμi-λi2,which is the mean number of customers in the G
e
o
/
G
e
o
/
1 queue with vacations in random environment.


## 6.1. TheG
e
o
/
G
/
1 Queue with Vacations

When the system is homogeneous, that is,λ
i
=
λ and μ
i
=
μ, the model translates into a regular G
e
o
/
G
/
1 queue with multiple vacations. From (23), P
(
z
) reduces to(47)Pz=θ1-ρ1-zSαz1-θ¯αzSαz-z,which is the PGF of the number of customers in the regular G
e
o
/
G
/
1 queue with multiple vacations, where S
(
z
) is the PGF of the service time distribution, α
(
z
)
=
λ
¯
+
λ
z. Expression (47) is in agreement with that in Tian and Zhang [3].


## 6.2. TheG
e
o
/
G
e
o
/
1 Queue with Vacations in Random Environment

Suppose that the service times in phasei, i
=
1,2
,
…
,
n, follow the geometric distribution with finite means 1
/
μ
i in the model; then S
i
(
z
)
=
μ
i
z
/
1
-
μ
¯
i
z in (23), and (26) yields(48)L=1-λ¯0θ¯θ¯θ2β+1-λ¯0θ¯θ2β∑i=1nqiμi-λiθ+λ0θ¯+θλiμ¯iμi-λi2,which is the mean number of customers in the G
e
o
/
G
e
o
/
1 queue with vacations in random environment.


## 7. Numerical Examples

To observe the effect of the different parameters on the main performance measures, we present some numerical examples in this section. We consider theG
e
o
/
G
e
o
/
1 queue with vacations in random environment, and we assume n
=
2; that is to say, the system has three phases. We set λ
0
=
0.3, λ
1
=
0.4, λ
2
=
0.5, μ
1
=
0.5, μ
2
=
0.8, q
1
=
0.4, q
2
=
0.6, and θ
=
0.3,0.5,0.7, unless they are considered as variables or their values are given in the respective figures.In Figure1, we study the influence of q
1 on L for different vacation rate θ. As intuition tells us, the mean number of customers in the system L increases with increasing values of q
1 for any θ. On the other hand, for a fixed q
1, L decreases with the increase of θ. Intuitively, as θ increases, the mean time of vacation decreases; then there are fewer customers in the system. And the behavior of L with respect to μ
1 is displayed in Figure 2. As to be expected, when μ
1 increases the mean number of customers in the system L decreases for any θ. If μ
1 is fixed, L decreases with the increase of θ.Figure 1
The effect ofq
1 on L.Figure 2
The effect ofμ
1 on L.The influence of arrival rateλ
0 on the probability that the system is empty P
0,0 is illustrated in Figure 3. It is observed that P
0,0 decreases as λ
0 increases, which also agrees with the intuitive expectations. In addition, for a fixed λ
0, P
0,0 increases with the increase of θ. Figure 4 describes the influence of μ
1 on the probability that the system is empty P
0,0. P
0,0 increases with the increase of μ
1, and, for a fixed μ
1, P
0,0 increases with the increase of θ, as we expected.Figure 3
The effect ofλ
0 on P
0,0.Figure 4
The effect ofμ
1 on P
0,0.


## 8. Conclusion

In this paper, we consider aG
e
o
/
G
/
1 queue with vacations in random environment. For this model, we derive the probability generating function of the number of customers in the system. Various system performance characteristics are obtained. And we show that the M
/
G
/
1 queue with vacations in random environment can be approximated by its discrete-time counterpart. We also perform some numerical examples to demonstrate the effect of various parameters on the performance characteristics.



---

*Source: 4029415-2016-06-15.xml*