# The Discrete-Time Bulk-ServiceGeo/Geo/1 Queue with Multiple Working Vacations

Authors: Jiang Cheng; Yinghui Tang; Miaomiao Yu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper deals with a discrete-time bulk-serviceGeo/Geo/1
queueing system with infinite buffer space and multiple working vacations. Considering an early arrival system, as soon as the server empties the system in a regular busy period, he leaves the system and takes a working vacation for a random duration at time n. The service times both in a working vacation and in a busy period and the vacation times are assumed to be geometrically distributed. By using embedded Markov chain approach and difference operator method, queue length of the whole system at random slots and the waiting time for an arriving customer are obtained. The queue length distributions of the outside observer’s observation epoch are investigated. Numerical experiment is performed to validate the analytical results.


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## Body

## 1. Introduction

Recently there has been a rapid increase in the literature on discrete-time queueing system with working vacations. These queueing models have been studied extensively and applied to computer networks, communication systems, and manufacturing systems. In the classical queueing system with server vacations, the server stops working during vacation periods. Suppose, however, that a system can be staffed with a substitute server during the times the main server is taking vacations. The service rate of the substitute server is different from (and probably lower than) that of the main server. This is the notion of working vacations recently introduced by Servi and Finn [1]. They studied an M/M/1 queue with multiple working vacations (M/M/1/WV). Their work is motivated by the analysis of a reconfigurable wavelength-division multiplexing (WDM) optical access network. In 2006, Wu and Takagi [2] generalized Servi and Finn’s M/M/1/WV queue to an M/G/1/WV queue. Baba [3] extended Wu and Takagi’s work to a renewal input GI/M/1 queue with working vacations and derived the steady-state system length distributions at an arrival and arbitrary epochs. The Geo/Geo/1 queue with single and multiple working vacations have been discussed in Li and Tian [4] and Tian et al. [5], respectively. Chae et al. [6] studied the GI/M/1 queue and GI/Geo/1 queue with single working vacation (SWV). The discrete-time infinite buffer GI/Geo/1 queue with multiple working vacations and vacation interruption has been studied in Li et al. [7, 8]. The discrete-time finite buffer GI/Geo/1 queue with multiple working vacations has been discussed by Goswami and Mund [9]. All the above studies on discrete-time single server queues have been carried out under the assumption that a server serves  singly at a time. However, there are many instances where the servers are carried out in batches to enhance the performance of the system. Over the last several years the discrete-time single server queues in batch service without vacations have been studied in Gupta and Goswami [10], Chaudhry and Chang [11], Alfa and He [12], and Yi et al. [13]. Lately, This type of queueing systems raise interest once more by many scholars such as Banerjee et al. [14, 15], Claeys et al. [16, 17].The continuous-time infinite buffer single server batch service queue with multiple vacations has been analyzed by Choi and Han [18], Chang and Takine [19]. The M/G/1 queue with bulk service and single vacation has been investigated by Sikdar and Gupta [20].As to the research to queueing systems with batch service and working vacations (or vacation), Yu et al. [21] considered a finite capacity and bulk-arrival and bulk-service continuous-time queueing system with multiple working vacations and partial batch rejection. Vijaya Laxmi and Yesuf [22] studied a renewal input infinite buffer batch service queue with single exponential working vacation and accessibility to batches. Goswami and Mund [23] investigated a discrete-time batch service renewal input queue with multiple working vacations. Note that although all the above studies on discrete-time bulk-service queues have been carried out, but, from their models, we can only obtain the average length of the waiting for service; the average length of the whole system cannot be obtained at prearrival, arbitrary and outside observer’s observation epochs. In fact, the number of customers being served in batches is a random variable; it is difficult to compute the average length of the whole system.This paper focuses on a discrete-time batch-service infinite bufferGeo/Geo/1 queueing system with multiple working vacations in which arrivals occur according to a geometrical input. By using embedded Markov chain approach, we obtained the operation rules of the one-step transition probability matrix, the average length at random slots, and the average waiting time for an arriving customer. This model has potential applications in computer networks where jobs are processed in batches.The rest of the paper is arranged as follows. In the next section, the model of the considered queueing system is described. In Section3, the stationary distribution of queue length at arbitrary slot epochs is discussed. In Section 4, we study the waiting time distribution. In Section 5, we discuss the outside observer’s observation epoch probabilities distribution. In Section 6, some numerical results and the sensitivity analysis of this system are given. Section 7 concludes this paper.


## 2. System Description

We consider a discrete-time bulk-service infinite buffer space queueing system with server multiple working vacations according to the rule of an early arrival system. Assume that the time axis is slotted into intervals of equal length with the length of a slot being unity, and it is marked as0,1,2,…,n,…, a potential arrival occurs in the interval (n,n+) and potential batch departures occur in (n-,n). In the meantime, the interarrival times T of customers are independent and geometrically distributed with probability mass function (p.m.f.) P{T=k}=pp-k-1,k≥1,p-=1-p.The customers are served in batches of variable capacity, the maximum service capacity for the server beinga(a≥1). Service times Sb during normal busy period and service times Sv during a working vacation are assumed independent and geometrically distributed with p.m.f. P{Sb=k}=μbμ-bk-1,k≥1,μ-b=1-μb and p.m.f. P{Sv=k}=μvμ-vk-1,k≥1,μ-v=1-μv, respectively. A new arriving customer cannot go into the queue being served immediately in spite of the working vacation period and the normal busy period. The server leaves the system and takes a working vacation at epoch n as soon as system becomes empty in regular period. The working vacation time follows a geometric distribution with parameter θ(0<θ<1) and its p.m.f. is P{V=k}=θθ-k-1,k≥1,θ-=1-θ. If there are some customers being served after the server finishes a working vacation, the service interrupted at the end of a vacation is lost and it is restarted with service rate μb at the beginning of the following service period, which means that the regular busy period starts. The various time epochs at which events occur are depicted in Figure 1.Figure 1
Various time epochs in EAS.We assume that interarrival times, service times, and working vacation times are mutually independent.In addition, the service discipline is first in first out (FIFO).  LetQn be the number of customers in the system at time n, and
(1)Jn={0,thesystemisinaworkingvacationperiodattimen,1,thesystemisinaregularbusyperiodattimen.
Then {Qn,Jn} is a Markov chain with the state space
(2)Ω={(0,0)}⋃{(k,j):k≥1,j=0,1},
where state (k,1),  k≥1 indicates that the system is in a regular busy period; state (k,0),  k≥0 indicates that the system is in a working vacation period and there k customers in the system.Using the lexicographical sequence for the states, the one-step transition block matrix can be written as(3)P=01234⋮aa+1⋮[A00BCDECFGECF0GECF00GE⋮⋮⋮⋱⋱⋱⋱CF0⋯00GE0HF0000GE⋮⋱⋱⋱⋱⋱⋱⋱⋱⋱],
where the first column in front of the matrix denotes the number of the customers for the first column block matrix in matrix P, and A00 is transition probability from (0,0) to (0,0); B is transition probability matrix from (0,0) to {(1,0),(1,1)}; C is transition probability matrix from {(i,0),(i,1)}(1≤i≤a) to (0,0); D is transition probability matrix from {(1,0),(1,1)} to {(1,0),(1,1)}; E is transition probability matrix from {(i,0),(i,1)}(i≥1) to {(i+1,0),(i+1,1)}(i≥1); F is transition probability matrix from {(i,0),(i,1)}(2≤i≤a) to {(1,0),(1,1)} and from {(i,0),(i,1)}(i≥a+1) to {(i-a+1,0),(i-a+1,1)}(i≥a+1); G is transition probability matrix from {(i,0),(i,1)}(i≥2) to {(i,0),(i,1)}(i≥2); H is transition probability matrix from {(i,0),(i,1)}(i≥a+1) to {(i-a,0),(i-a,1)}(i≥a+1).We have(4)A00=p-+pθ-μv,B=[pθ-μ-vpθ],C=[p-θ-μvp-μb],D=[p-θ-μ-v+pθ-μvp-θ0p-μ-b+pμb],E=[pθ-μ-vpθ0pμ-b],F=[pθ-μv00pμb],G=[p-θ-μ-vp-θ0p-μ-b],H=[p-θ-μv00p-μb].


## 3. The Stationary Queue Size Distributions at Random Slots

Assume that(Q,J) is the stationary limit of {Qn,Jn}, and its distribution is denoted as πk,j=limn→∞P{Qn=k,Jn=j}=P{Q=k,J=j},  (k,j)∈Ω, we have the following theorem.Theorem 1.
Ifρ=p/aμb<1, ρ0=p/aμv<1, πk,j(k≥1,j=0,1) are given by
(5)πk,0=pθ-μ-v(1-ξ)ξk-1ωπ0,0,πk,1=c0′rk+pθθ-μ-v(p+p-ξ)(1-ξ)ξk-1π0,0ωϖ,(k≥1),
where
(6)ω=1-θ-[1-p-μv]-p-θ-μvξa,ϖ=[(1-p-μ-b)ξ-(pμ-b+pμbξa+p-μbξa+1)],c0′=(pθ{∫ωϖ+θ-μ-v×{pϖ(1-ξ)+(p+p-ξ)[ϖ-p-μb(1-ξa)]∫}}×(1-r)∫)×(rωϖp-μb(1-ra))-1π0,0,π0,0={∫1+(pθ-μ-vϖ+pθθ-μ-v(p+p-ξ))(ωϖ)-1+(pθ{ωϖ+θ-μ-v×{pϖ(1-ξ)+(p+p-ξ)[ϖ-p-μb(1-ξa)]}ωϖ+θ-μ-v}pθ{ωϖ+θ-μ-v)×(ωϖp-μb(1-ra))-1∫}-1,0<ξ<1 and ξ is the root of the equation p-μvθ-za+1+pμvθ-za-(1-p-μ-vθ-)z+pμ-vθ-=0,0<r<1 and r is the root of the equation p-μbza+1+pμbza+pμ-b-(1-p-μ-b)z=0.Proof.
According to the one-step transition probability matrix, we can see which is notQBD process, the method of matrix-geometric solution is invalid. Based on the stationary equations obtained directly by stochastic balance, we have
(7)π0,0=π0,0A00+∑i=1a(πi,0,πi,1)C,(π1,0,π1,1)=π0,0B+(π1,0,π1,1)D+∑i=2a(πi,0,πi,1)F+(πa+1,0,πa+1,1)H,(πk,0,πk,1)=(πk-1,0,πk-1,1)E+(πk,0,πk,1)G+(πa+k-1,0,πa+k-1,1)F+(πa+k,0,πa+k,1)H,k≥2.
Then, we obtain the following equations:
(8)π0,0=π0,0(p-+pθ-μv)+p-θ-μv∑i=1aπi,0+p-μb∑i=1aπi,1,(9)π1,0=pθ-μ-vπ0,0+p-θ-μ-vπ1,0+pθ-μv∑i=1aπi,0+p-θ-μvπa+1,0,(10)πk,0=pθ-μ-vπk-1,0+p-θ-μ-vπk,0+pθ-μvπa+k-1,0+p-θ-μvπa+k,0,k≥2,(11)π1,1=pθπ0,0+p-θπ1,0+p-μ-bπ1,1+pμb∑i=1aπi,1+p-μbπa+1,1,(12)πk,1=pθπk-1,0+pμ-bπk-1,1+p-θπk,0+p-μ-bπk,1+pμbπa+k-1,1+p-μbπa+k,1,k≥2.
According to the characteristic of difference equations, let πk+j,0=Ejπk,0 [24], j∈Z;k=0,1,2,…, where E denotes the difference operator for the difference equation, substituting it into (10), (10) can be written as
(13)(p-θ-μvEa+1+pθ-μvEa-(1-p-θ-μ-v)E+pθ-μ-v)πk-1,0=0.
The auxiliary equation is given by
(14)p-θ-μvza+1+pθ-μvza-(1-p-θ-μ-v)z+pθ-μ-v=0.
Let f(z)=p-θ-μvza+1+pθ-μvza+p-θ-μ-vz+pθ-μ-v and g(z)=-z. Using Rouché's theorem, it can be shown that there is only one zero real root falls in the unit circle (Note: the root must be real root; otherwise, there are two roots at least fall in the unit circle, because the imaginary roots of an equation appear in pairs.) We denote this root by ξ(0<ξ<1) and the other a roots by ξi, |ξi|≥1(i=1,2,3,…,a). So ξ satisfies f(ξ)+g(ξ)=0. Therefore, the solution of (10) can be written as
(15)πk,0=c0ξk+∑i=1aciξik,k≥1.
Since ci(i=1,2,3,…,a)=0 (otherwise, the probability πk,0 tends to ∞ when k tends to ∞), we get πk,0=c0ξk(k≥1).
Letk=1, we get c0=ξ-1π1,0, then
(16)πk,0=π1,0ξk-1,k≥1.
Substituting (16) into (9), we obtain
(17)π1,0=pθ-μ-v(1-ξ)ωπ0,0,
where ω=1-θ-[1-p-μv]-p-θ-μvξa.
Hence,(18)πk,0=pθ-μ-v(1-ξ)ξk-1ωπ0,0,k≥1.
For k≥2, the difference equation (12) can be written as
(19)pμ-bπk-1,1+(p-μ-b-1)πk,1+pμbπa+k-1,1+p-μbπa+k,1+pθθ-μ-v(p+p-ξ)(1-ξ)ξk-2ωπ0,0=0.
Using πk+j,1=Ejπk,1, j∈Z; k=1,2,…, the auxiliary equation of (19) such that
(20)p-μbza+1+pμbza-(1-p-μ-b)z+pμ-b=0.
Let G(z)=p-μbza+1+pμbza+p-μ-bz+pμ-b, obviously G(1)=1, G′(1)=(a+1)p-μb+apμb+p-μ-b=aμb+1-p. Since ρ=p/(aμb)<1, that is, p<aμb, we can see that G′(1)>1. According to Hunter [25], the equation z=G(z) has a unique real root in unit circle, which can be denoted by r; the other a roots can be denoted by ri, |ri|≥1(i=1,2,…,a). As mentioned above, the general solution of (20) can be written as
(21)z*=c0′rk,k≥1.
Using the method of nonhomogeneous difference operator, the special solution of (19) can be written as
(22)πk,1*=pθθ-μ-v(p+p-ξ)(1-ξ)ξk-1π0,0ωϖ,k≥1,
where ϖ=[(1-p-μ-b)ξ-(pμ-b+pμbξa+p-μbξa+1)].
From (21) and (22), we have
(23)πk,1=c0′rk+pθθ-μ-v(p+p-ξ)(1-ξ)ξk-1π0,0ωϖ,k≥1.
Substituting (17), (18), and (23) into (11), we obtain
(24)c0′=(∫pθ{ωϖ+θ-μ-v{pϖ(1-ξ)+(p+p-ξ)[ϖ-p-μb(1-ξa)]}}×(1-r)∫)(rωϖp-μb(1-ra))-1π0,0.
Substituting (24) into (23), we can obtain πk,1(k≥1).
Using∑i=0∞πi,0+∑i=1∞πi,1=1, we can get
(25)π0,0={∫1+(pθ-μ-vϖ+pθθ-μ-v(p+p-ξ))(ωϖ)-1+(pθ{∫ωϖ+θ-μ-v×{pϖ(1-ξ)+(p+p-ξ)[ϖ-p-μb(1-ξa)]}∫})×(ωϖp-μb(1-ra))-1∫}-1.Remark 2.
Ifθ→0 and a=1, this queueing system is equivalent to Geom/Geom/1 queueing system with a server serves customers singly. Because of ρ0=p/aμv<1 and 1≤a, then p/μv<1. Hence, 0<ξ=(pμ-v/p-μv)<1. We have
(26)π0,0=1-ξ,πk,0=(1-ξ)ξk,k≥1,
which are matched with the results given by Tian and Ma [26].Corollary 3.
DefineJ as the state of system at random slots, the steady-state probability of this system at random slots can be written as
(27)P{J=0}=(1+pθ-μ-vω)π0,0,P{J=1}={(pθ{ωϖ+θ-μ-v×{pϖ(1-ξ)+(p+p-ξ)[ϖ-p-μb(1-ξa)]}pθ{ωϖ+θ-μ-v}(pθ{ωϖ+θ-μ-v)×(ωϖp-μb(1-ra))-1+(pθθ-μ-v(p+p-ξ))(ωϖ)-1}π0,0.Theorem 4.
If|z|≤1, the probability generating function (PGF) of steady-state queue length at random slots is given by
(28)L(z)={1+{c0′rz1-rz+pθ-μ-v(1-ξ)zω(1-ξz)+pθθ-μ-v(p+p-ξ)(1-ξ)zωϖ(1-ξz)}}π0,0,
and the average queue length is
(29)E(L)={rc0′(1-r)2+[ϖ+θ(p+p-ξ)]pθ-μ-vωϖ(1-ξ)}π0,0.Proof.
In the steady state, the queue lengthL at random slots has marginal distribution as
(30)P{L=0}=π0,0,P{L=k}=πk,0+πk,1=c0′rk+pθ-μ-v(1-ξ)ξk-1ωπ0,0+pθθ-μ-v(p+p-ξ)(1-ξ)ξk-1π0,0ωϖ,(k≥1).
Using L(z)=P{L=0}+∑k=1∞P{L=k}zk, we can obtain (28) easily; furthermore, taking derivation to L(z) and let z=1, we can get (29).


## 4. The Waiting Time Distribution

Let the random variableTq be the total waiting time of an arriving customer in the queue, Nw represents the number of the customers in the system. Assume that an arriving customer finds i customers in the system, the conditional distribution law that he waits for k slots is subject to wi(k)=P{Tq=k/Nw=i}, i=0,1,2,…,k=0,1,2,…, and PGF is Wi(z)=∑k=0∞wi(k)zk. In the steady state, the waiting time with finite mean wq has PGF wq(z)=∑i=0∞πilWi(z),l=0,1.Theorem 5.
In the steady state, the PGF of waiting time for an arriving customer is given by(31)wq(z)=π0,0+pθθ-μ-v(p+p-ξ)(1-ξa)q(z)ωϖπ0,0+pμ-v(1-ξa)q~(θ-z)ωπ0,0+pθθ-μ-v(p+p-ξ){1-q(z)ξa-[1-q(z)]ξa-1}ξaq(z)ωϖ[1-q(z)ξa]π0,0+{1-q~(θ-z)ξa-[1-q~(θ-z)]ξa-1}pμ-vq~(θ-z)ξaω[1-q~(θ-z)ξa]π0,0+c0′{r-r2a1-r+q(z)r2a(1-ra)(1-r)[1-q(z)ra]}q(z)+pθμ-vq(z)(1-ξa)ξa-1ω(1-μ-vzθ-)[1-ξaq(z)][1-ξaq~(zθ-)]π0,0,
and the average waiting time as
(32)wq=1μb{(1-ra+r2a-1)r(1-ra)(1-r)c0′+pθθ-μ-v(p+p-ξ)(1-ξa+ξ2a-1)ωϖ(1-ξa)π0,0+pθμ-vξa-1ω(1-μ-vθ--μvθ-ξa)(1-ξa)π0,0}+pμvμ-vθ-ω×{+(1-ξa)((1-μ-vθ-)2)-1({[1+ξa-1+θ(ξa-1-ξ2a-1)-2ξa]×(1-μ-vθ-)-μvθ-(ξa-1-ξa)ξa}ξa)×((1-μ-vθ-)(1-μ-vθ--μvθ-ξa)2)-1+(1-ξa)((1-μ-vθ-)2)-1}π0,0+pθθ-μ-v2ξa-1ω(1-μ-vθ-)(1-μ-vθ--μvθ-ξa)π0,0.Proof.
Firstly, we define⌊x⌋ as a greatest integer function (floor), which returns the greatest integer less than or equal to a real number x. An arriving customer may observe the system in any one of the following two cases.
Case 1. When Tq=0, this case has no customers in the system and the server is on vacation, that is,
(33)w0(0)=P{Tq=0Nw=0}=π0,0.
Case 2. When Tq=m,(m≥1), there are two cases as follows.
(1) The server is on a normal busy period andi customers in the system, meantime an arriving customer cannot go into the queue being served immediately.
Under this condition, an arriving customer has to wait for one period of service for1≤i≤a and ⌊i/a⌋ periods of service for i>a. Each period of service Sbi(i=1,2,…) is independent and geometrically distributed with p.m.f.  P{Sbi=k}=μbμ-bk-1, k≥1,μ-b=1-μb, which has PGF as μbz/(1-μ-bz).
We have(34)wi(m)={P{sb1=mNw=i},i≤a,P{sb1+sb2+⋯+sb⌊i/a⌋=mNw=i},i>a.
Hence,
(35)Wi(z)=∑m=1∞wi(m)zm=μbz1-μ-bz,1≤i≤a,Wi(z)=∑m=1∞wi(m)zm=(μbz1-μ-bz)⌊i/a⌋,i>a.
Let q(z)=ubz/(1-μ-bz), the PGF of the waiting time is given by
(36)pθθ-μ-v(p+p-ξ)(1-ξa)q(z)ωϖπ0,0+c0′{r-r2a1-r+q(z)r2a(1-ra)(1-r)[1-q(z)ra]}q(z)+pθθ-μ-v(p+p-ξ){1-q(z)ξa-[1-q(z)]ξa-1}ξaq(z)ωϖ[1-q(z)ξa]π0,0.
(2) An arriving customer finds the server is on vacation.
Letsvj be the jth length of the period of service with service rate μv and let sv(j) be the sum of the length of j periods of service with service rate μv, j=1,2,3,…, where sv(0)=0. Each period of service Svi(i=1,2,…) is mutually independent and geometrically distributed with p.m.f. P{Svi=k}=μvμ-vk-1,  k≥1, μ-v=1-μv. There are two cases in this condition.
(A) A vacation is going on whereas all of the arrived customers have been served. Then, an arriving customer has to wait for one period of service fori≤a and ⌊i/a⌋ periods of service for i>a. We have
(37)wi(m)=P{sv1=mNw=i;V≥m},1≤i≤a,Wi(z)=∑m=1∞wi(m)zm=μvz1-θ-μ-vz,1≤i≤a,wi(m)=P{sv1+sv2+⋯+sv⌊i/a⌋=mNw=i;V≥m},i>a,Wi(z)=∑m=1∞wi(m)zm=1θ-(μvθ-z1-μ-vθ-z)⌊i/a⌋,i>a.
Letq~(z)=μvz/(1-μ-vz). The PGF of the waiting time is given by
(38)pμ-v(1-ξa)ωq~(θ-z)π0,0+{1-q~(θ-z)ξa-[1-q~(θ-z)]ξa-1}pμ-vq~(θ-z)ξaω[1-q~(θ-z)ξa]π0,0.
(B) If a vacation is over andj(j=0,1≤i≤a;1≤j<⌊i/a⌋,i>a) periods of service ended, the service rate is converted to μb from μv and the normal busy period begins. The waiting time of an arriving customer should be equal to the sum of the server’s vacation times and one period of service for 1≤i≤a and ⌊i/a⌋-j periods of service for i>a. The service rate of ⌊i/a⌋-j periods of service is μb. We have
(39)wi(m)=P{Tq=mN=i;V<sv1},1≤i≤a,wi(m)=∑j=0⌊i/a⌋-1P{Tq=mN=i;sv(j)≤V<sv(j+1)},i>a.
Hence,
(40)wi(m)=∑j=0⌊i/a⌋-1P{Tq=mN=i;sv(j)≤V<sv(j+1)}=∑j=0⌊i/a⌋-1P{V+sb1+sb2+⋯+sb⌊i/a⌋-j=m;sv(j)≤V<sv(j+1)}=∑j=0⌊i/a⌋-1∑u=1m-⌊i/a⌋+jP{V=u}×P{sb1+sb2+⋯+sb⌊i/a⌋-j=m-u}×P{sv(j)≤u<sv(j+1)},i>a.
The PGF of the waiting time can be given by
(41)pθμ-vq(z)(1-ξa)ξa-1ω(1-μ-vzθ-)[1-ξaq(z)][1-ξaq~(zθ-)]π0,0.
Adding (33)–(41), we can get (31), using (dwq(z)/dz)|z=1, we can obtain (32).


## 5. Outside Observer’s Observation Epoch Distributions

For an early arrive system, since an outside observer’s observation epoch falls in the time interval after a potential arrival and before a potential batch departure, let,π~n,0, π~n,1 be n(n≥0) customers in the system and the server is on vacation (including the servicing customers), n customers in the system and the server is in regular busy period (including the servicing customers). Through observing the relationship between random slot t and the outside observer’s observation epoch (*), we have
(42)π~0,0=p-π0,0,π~n,0=p-θ-πn,0+pθ-πn-1,0,(n≥1),π~1,1=p-π1,1+pθπ0,0+p-θπ1,0,π~n,1=p-πn,1+pπn-1,1+p-θπn,0+pθπn-1,0,(n≥2).


## 6. Numerical Results and the Sensitivity Analysis of this System

In this section, we present some numerical results in self-explanatory tables and graphs for queue length distributions at random slots and all the numerical results have been obtained using the results derived in this paper.We observe thatπn,0 and π~n,0 monotonically increase whereas πn,1 and π~n,1 monotonically decrease as a increases in Tables 1–4. This situation continues until a is equal to some constant; all data will tend to be a steady state. The above description is consistent with actual situation. In the meantime, E(L) and E(wq) monotonically decease as a increases. In Figures 2 and 3, fixing a=10, p=0.3, μb=0.5, and θ=0.3,0.5,0.7, we have plotted the effect of various vacation service rates on the average queue length and the average waiting time, respectively. We observe that the average queue length and the average waiting time decrease as the vacation service rate increases. In Figure 4, fixing p=0.3,μv=0.4,μb=0.6, and θ=0.7, the steady-state average queue length equals 0.2597 from a=4 on and the steady-state average waiting time equals 0.8421 from a=5 on. They do not change as the batch size increases.Table 1
Queue length distributions at random slots fora=1, p=0.3, μv=0.4, μb=0.6, and θ=0.7.
n
π
n
,
0
π
n
,
1
n
π
~
n
,
0
π
~
n
,
1
0
0.5580
0
0
0.3906
0
1
0.0363
0.3128
1
0.0603
0.3539
2
0.0022
0.0679
2
0.0037
0.1501
3
1.39
E
-
04
0.0128
3
2.32
E
-
04
0.0299
4
8.64
E
-
06
0.0023
4
1.44
E
-
05
0.0055
5
5.35
E
-
07
4.17
E
-
04
5
8.90
E
-
07
9.92
E
-
04
6
3.32
E
-
08
7.43
E
-
05
6
5.51
E
-
08
1.77
E
-
04
7
2.05
E
-
09
1.32
E
-
05
7
3.42
E
-
09
3.16
E
-
05
8
1.27
E
-
10
2.35
E
-
06
8
2.12
E
-
10
5.62
E
-
06
9
7.89
E
-
12
4.19
E
-
07
9
1.31
E
-
11
1.00
E
-
06
10
4.89
E
-
13
7.45
E
-
08
10
8.13
E
-
13
1.78
E
-
07
11
3.03
E
-
14
1.32
E
-
08
11
5.04
E
-
14
3.16
E
-
08
12
1.88
E
-
15
2.36
E
-
09
12
3.12
E
-
15
5.62
E
-
09
13
1.16
E
-
16
4.19
E
-
10
13
1.93
E
-
16
1.00
E
-
09
14
7.21
E
-
18
7.45
E
-
11
14
1.20
E
-
17
1.78
E
-
10
15
4.47
E
-
19
1.33
E
-
11
15
7.43
E
-
19
3.16
E
-
11
Sum
1
Sum
1
E
(
L
)
=
0.2741, E(wq)=1.2783.Table 2
Queue length distributions at random slots fora=4, p=0.3, μv=0.4, μb=0.6, and θ=0.7.
n
π
n
,
0
π
n
,
1
n
π
~
n
,
0
π
~
n
,
1
0
0.6092
0
0
0.4265
0
1
0.0394
0.263
1
0.0654
0.3313
2
0.0024
0.0623
2
0.0041
0.1319
3
1.51
E
-
04
0.012
3
2.51
E
-
04
0.0277
4
9.37
E
-
06
0.0022
4
1.56
E
-
05
0.0052
5
5.80
E
-
07
3.95
E
-
04
5
9.65
E
-
07
9.39
E
-
04
6
3.60
E
-
08
7.05
E
-
05
6
5.98
E
-
08
1.68
E
-
04
7
2.23
E
-
09
1.26
E
-
05
7
3.70
E
-
09
2.99
E
-
05
8
1.38
E
-
10
2.23
E
-
06
8
2.30
E
-
10
5.33
E
-
06
9
8.56
E
-
12
3.97
E
-
07
9
1.42
E
-
11
9.48
E
-
07
10
5.30
E
-
13
7.07
E
-
08
10
8.81
E
-
13
1.69
E
-
07
11
3.29
E
-
14
1.26
E
-
08
11
5.46
E
-
14
3.00
E
-
08
12
2.04
E
-
15
2.24
E
-
09
12
3.38
E
-
15
5.33
E
-
09
13
1.26
E
-
16
3.98
E
-
10
13
2.10
E
-
16
9.49
E
-
10
14
7.82
E
-
18
7.07
E
-
11
14
1.30
E
-
17
1.69
E
-
10
15
4.84
E
-
19
1.26
E
-
11
15
8.05
E
-
19
3.00
E
-
11
Sum
1
Sum
1
E
(
L
)
=
0.2597, E(wq)=0.8425.Table 3
Queue length distributions at random slots fora=5, p=0.3, μv=0.4, μb=0.6, and θ=0.7.
n
π
n
,
0
π
n
,
1
n
π
~
n
,
0
π
~
n
,
1
0
0.6095
0
0
0.4266
0
1
0.0394
0.2627
1
0.0655
0.3312
2
0.0024
0.0622
2
0.0041
0.1319
3
1.51
E
-
04
0.012
3
2.51
E
-
04
0.0277
4
9.37
E
-
06
0.0022
4
1.56
E
-
05
0.0052
5
5.81
E
-
07
3.95
E
-
04
5
9.65
E
-
07
9.39
E
-
04
6
3.60
E
-
08
7.05
E
-
05
6
5.98
E
-
08
1.68
E
-
04
7
2.23
E
-
09
1.25
E
-
05
7
3.71
E
-
09
2.99
E
-
05
8
1.38
E
-
10
2.23
E
-
06
8
2.30
E
-
10
5.33
E
-
06
9
8.56
E
-
12
3.97
E
-
07
9
1.42
E
-
11
9.48
E
-
07
10
5.30
E
-
13
7.06
E
-
08
10
8.82
E
-
13
1.69
E
-
07
11
3.29
E
-
14
1.26
E
-
08
11
5.46
E
-
14
3.00
E
-
08
12
2.04
E
-
15
2.23
E
-
09
12
3.39
E
-
15
5.33
E
-
09
13
1.26
E
-
16
3.97
E
-
10
13
2.10
E
-
16
9.48
E
-
10
14
7.82
E
-
18
7.07
E
-
11
14
1.30
E
-
17
1.69
E
-
10
15
4.85
E
-
19
1.26
E
-
11
15
8.06
E
-
19
3.00
E
-
11
Sum
1
Sum
1
E
(
L
)
=
0.2597, E(wq)=0.8421.Table 4
Queue length distributions at random slots fora=10, p=0.3, μv=0.4, μb=0.6, and θ=0.7.
n
π
n
,
0
π
n
,
1
n
π
~
n
,
0
π
~
n
,
1
0
0.6095
0
0
0.4267
0
1
0.0394
0.2627
1
0.0655
0.3312
2
0.0024
0.0622
2
0.0041
0.1318
3
1.51
E
-
04
0.012
3
2.51
E
-
04
0.0277
4
9.37
E
-
06
0.0022
4
1.56
E
-
05
0.0052
5
5.81
E
-
07
3.95
E
-
04
5
9.65
E
-
07
9.38
E
-
04
6
3.60
E
-
08
7.05
E
-
05
6
5.98
E
-
08
1.68
E
-
04
7
2.23
E
-
09
1.25
E
-
05
7
3.71
E
-
09
2.99
E
-
05
8
1.38
E
-
10
2.23
E
-
06
8
2.30
E
-
10
5.33
E
-
06
9
8.56
E
-
12
3.97
E
-
07
9
1.42
E
-
11
9.48
E
-
07
10
5.30
E
-
13
7.06
E
-
08
10
8.82
E
-
13
1.69
E
-
07
11
3.29
E
-
14
1.26
E
-
08
11
5.46
E
-
14
3.00
E
-
08
12
2.04
E
-
15
2.23
E
-
09
12
3.39
E
-
15
5.33
E
-
09
13
1.26
E
-
16
3.97
E
-
10
13
2.10
E
-
16
9.48
E
-
10
14
7.82
E
-
18
7.07
E
-
11
14
1.30
E
-
17
1.69
E
-
10
15
4.85
E
-
19
1.26
E
-
11
15
8.06
E
-
19
3.00
E
-
11
Sum
1
Sum
1
E
(
L
)
=
0.2597, E(wq)=0.8421.Figure 2
Effect ofμv on the average queue length.Figure 3
Effect ofμv on the average waiting time.Figure 4
Effect ofa on the average queue length and the average waiting time.


## 7. Conclusions

AGeom/Geom[a]/1/MWV queueing system has been investigated. Assume that the server takes a working vacation after emptying the system in regular busy period. By using embedded Markov chain approach and the method of nonhomogeneous and homogeneous difference operator, the number of customers of the whole system at random slots has been discussed. This is different from general batch service queue literatures (excluding customers being served). The waiting time for an arriving customer and numerical results are obtained. In the future, further study such as Geom[a]/Geom[b]/1/MWV queue will be the research topic using similar idea and method.



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*Source: 587269-2013-02-25.xml*