# The Steady-State System Size Distribution for a Modified D-Policy Geo/G/1 Queueing System

Authors: Renbin Liu; Zhaohui Deng
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

This paper examines a discrete-time modified
D-policy Geo/G/1 queue with Bernoulli feedback. Using a
decomposition method, the steady-state system size distribution at
epochn+ is obtained. Moreover, the steady-state system size
distributions at epochs n- and n are also derived. Two
special cases are given. Finally, a wireless local area network is
numerically presented to validate the applicability of steady-state
system size distribution and its important application in system
capacity design.


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## Body

## 1. Introduction

The D-policy of queueing systems has been extensively studied in lots of research papers for the past decades. In contrast with the N-policy of Yadin and Naor [1] and the T-policy of Heyman [2], the D-policy controls the queueing system by the workload of waiting customers (i.e., the server resumes its service only when the sum of the service times of all waiting customers exceeds a positive constant D). The D-policy can be used in many real-world queueing systems to control the number of server startup and thereby reduces the average cost of system.The D-policy M/G/1 queue was first proposed by Balachandran [3] and Balachandran and Tijms [4]. Later on, this kind of queue was discussed by Boxma [5], Tijms [6], Artalejo [7], Gakis et al. [8], and Rhee [9]. The other works on the continuous-time D-policy queues can be found in [10–15] and references therein. For related literature on discrete-time D-policy queues the reader may refer to [16]. Recently, Lee et al. [17] analyzed the discrete-time MAP/G/1 queue under the D-policy and derived the probability generating functions of the queue length, workload, waiting time, and sojourn time distributions. Wei et al. [18] dealt with the transient and equilibrium properties of the queue length for a discrete-time Geo/G/1 queueing system with delayed D-policy and gave the stochastic decomposition of the queue length in equilibrium. However, compared with the continuous-time counterparts, the discrete-time D-policy queueing systems received less attention in the literature.The study of feedback queues was initiated by Takács [19]. The papers of Choi et al. [20], Choi and Kim [21], and Kumar and Raja [22] and their references provided us a wide study of different continuous-time feedback queues. Atencia and Moreno [23] first introduced a discrete-time queue with feedback. The phenomenon of feedback in the queueing systems occurs in many practical situations. For instance, in telecommunication systems, when messages are received with errors at the destination, they are resent. In a call center, users may repeat calls when they do not completely solve their problems and so forth. To the best of our knowledge, the existing literature on feedback queues focuses mainly on the continuous case. Little work has appeared in the discrete-time research area.In spite of the importance of D-policy and feedback in queueing models, there is only very little literature concerned with D-policy queues with feedback. Moreover, as far as we know, there is no work about the discrete-time D-policy queues with feedback. On the other hand, the existing D-policy queues focus on the probability generating function (PGF) of steady-state system size rather than the steady-state system size distribution. The first objective in this paper is to introduce a discrete-time modified D-policy Geo/G/1 queueing system with Bernoulli feedback, in whichD is a fixed positive integer value. Next, in such a queue, we will obtain the steady-state system size distributions at epochs n+, n-, and n, where n- and n+ represent lim⁡Δt→0(n-|Δt|) and lim⁡Δt→0(n+|Δt|), respectively. Lastly, we numerically present the applicability and application of steady-state system size distributions.Our queueing system can be used to model many practical situations. For example, in a wireless local area network (WLAN), the access point (AP) is a specially configured node on WLAN and acts as a central receiver and transmitter of WLAN radio signals. Once the radio signals are sent to the AP, they will be identified and transmitted if the AP is free. When the AP is busy, the received radio signals are placed in queue. Because of power saving, the AP is designed to start serving exhaustively whenever the workload of radio signals reaches a predetermined level. When a radio signal is transmitted with errors at the destination, it is retransmitted. In view of the fact that the receptions and transmissions of radio signals are proceeded in fixed time intervals, this kind of WLAN can be modeled as a discrete-time modified D-policy queue with feedbacks. Moreover, for the above WLAN, the steady-state system size distribution obtained in Section3 is very useful in network capacity design (see Section 5).This paper is organized as follows. The next section presents assumptions and preliminaries. Section3 analyzes the PGF of transient-state system size distribution. Based on this PGF the steady-state system size distributions at epochs n+, n-, and n are obtained. Sections 4 and 5 give two special cases and numerically illustrate the applicability and important application of steady-state system size distribution for a WLAN. Conclusions are drawn in Section 6.


## 2. Queue Assumptions and Some Preliminaries

We consider a discrete-time modified D-policy Geo/G/1 queueing system with Bernoulli feedback where the time is divided into constant length intervals (called slots). It is well known that the probability of an arrival and a departure occurring simultaneously is not zero in discrete time. This probability is positive in the discrete-time setting. That is why the order of the arrivals and departures must be stated. Usually, there are two models in discrete-time queues, that is, late arrival system with delayed access (LAS-DA) and early arrival system (EAS), which are also known as arrival first (AF) and departure first (DF) policies, respectively. These concepts and detailed discussions can be found in Hunter [24]. Here, we only discuss the mentioned model with LAS-DA; that is, a potential arrival can only take place in (n-,n),n=0,1,2,…, and a potential departure or feedback can only take place in (n,n+),n=1,2,…. We do not permit a departure at such time point for a customer who has just arrived the instant previously to an empty system. Further, we suppose that no customer arrives in the beginning time interval (0-,0) and no departure occurs in (0,0+).The customer arrival intervals,{τi,i≥1}, are independent identically distributed (i.i.d) random variables following a geometric distribution P{τi=j}=λ(1-λ)j-1(j≥1,0<λ<1). When the sum of the service times of all waiting customers exceeds a fixed positive integer value D, the server restarts service in the first-come, first-served (FCFS) order until the system becomes empty. The service times, denoted by {χn,n≥1}, are i.i.d random variables obeying an arbitrary distribution gj=P{χn=j},j≥1, with probability generating function (PGF) G(z)=∑j=1∞‍gjzj, |z|<1 and mean μ. The customer whose service has been just completed leaves forever with probability θ(0<θ≤1) or returns to the queue with complementary probability 1-θ for the next service. At initial epoch 0+, the server is idle and then begins to serve the first arrival if the number of customers in the system N(0+)=0 or immediately begins to serve if N(0+)>0. After the first busy period, the server will take a modified D-policy. All random variables involved in the system are independent of one another.For later discussion, some preliminaries are presented as follows.


### 2.1. Total Service Time of One Customer before Departure

Letχni, χ^n, and Ωn denote the ith service time, total service time, and total service number of customer n before departure, respectively. Then according to queue assumptions we get P(Ωn=k)=θ(1-θ)k-1,k≥1, and χ^n=∑i=1Ωn‍χni,n≥1 are i.i.d. random variables, where χni,i,n≥1 are mutually independent and have the same distribution as χn,n≥1.Further, the probability distributiong^j, PGF G^(z), and mean E(χ^) of total service time χ^ for one customer before departure are given, respectively, by
(1)g^j=P(χ^=j)=P{∑i=1Ωn‍χni=j}=∑k=1j‍θ(1-θ)k-1P{∑i=1k‍χni=j},nnnnnnnnnnnnnnnnnnnnj≥1,G^(z)=∑j=1∞‍g^jzj=θG(z)1-(1-θ)G(z),|z|<1,E(χ^)=dG^(z)dz|z=1=μθ.


### 2.2. The System Idle Period

It denotes the time length from the time when the system becomes empty until the first customer arrives and enters the system.LetIi denote the ith system idle period initiated with the initial state N(0+)(≥0); then it follows from the queue assumptions that {Ii,i≥1} are independent random variables following an identical geometric distribution with parameter λ.


### 2.3. The Server Busy Period

It denotes the time length from the time when the server begins to serve until the system becomes empty.Denote byb the server busy period initiated with one customer and its PGF is given by b(z)=∑j=0∞‍P(b=j)zj, |z|<1; then similar to the analysis in [25], b(z) is the root of the equation b(z)=G^[(1-λ)z+λb(z)z], and the mean is
(2)E(b)={μ(θ-λμ),ρ<1,∞,ρ≥1,
where ρ=λμ/θ represents the traffic intensity of the considered queue.Letb〈i〉 denote the server busy period initiated with i customers; then b〈i〉 can be expressed as b〈i〉=ς1+⋯+ςi, where ς1,…,ςi are mutually independent with the same distribution as b. So the PGF of b〈i〉 is given by bi(z), |z|<1.


## 2.1. Total Service Time of One Customer before Departure

Letχni, χ^n, and Ωn denote the ith service time, total service time, and total service number of customer n before departure, respectively. Then according to queue assumptions we get P(Ωn=k)=θ(1-θ)k-1,k≥1, and χ^n=∑i=1Ωn‍χni,n≥1 are i.i.d. random variables, where χni,i,n≥1 are mutually independent and have the same distribution as χn,n≥1.Further, the probability distributiong^j, PGF G^(z), and mean E(χ^) of total service time χ^ for one customer before departure are given, respectively, by
(1)g^j=P(χ^=j)=P{∑i=1Ωn‍χni=j}=∑k=1j‍θ(1-θ)k-1P{∑i=1k‍χni=j},nnnnnnnnnnnnnnnnnnnnj≥1,G^(z)=∑j=1∞‍g^jzj=θG(z)1-(1-θ)G(z),|z|<1,E(χ^)=dG^(z)dz|z=1=μθ.


## 2.2. The System Idle Period

It denotes the time length from the time when the system becomes empty until the first customer arrives and enters the system.LetIi denote the ith system idle period initiated with the initial state N(0+)(≥0); then it follows from the queue assumptions that {Ii,i≥1} are independent random variables following an identical geometric distribution with parameter λ.


## 2.3. The Server Busy Period

It denotes the time length from the time when the server begins to serve until the system becomes empty.Denote byb the server busy period initiated with one customer and its PGF is given by b(z)=∑j=0∞‍P(b=j)zj, |z|<1; then similar to the analysis in [25], b(z) is the root of the equation b(z)=G^[(1-λ)z+λb(z)z], and the mean is
(2)E(b)={μ(θ-λμ),ρ<1,∞,ρ≥1,
where ρ=λμ/θ represents the traffic intensity of the considered queue.Letb〈i〉 denote the server busy period initiated with i customers; then b〈i〉 can be expressed as b〈i〉=ς1+⋯+ςi, where ς1,…,ςi are mutually independent with the same distribution as b. So the PGF of b〈i〉 is given by bi(z), |z|<1.


## 3. The Analysis of System Size Distributions

In this section, by a decomposition method, we first analyze the PGF of transient system size distribution. Then based on this PGF we obtain the steady-state system size distribution at epochn+. Lastly we derive the steady-state system size distributions at epochs n- and n.


### 3.1. PGF for Transient System Size Distribution at Epochn+ in Server Busy Period b

Denote byQj(n+)=P{b>n+,N(n+)=j},j≥1 transient system size distribution at epoch n+ in server busy period b, and its PGF is qj(z)=∑n=0∞‍Qj(n+)zn, |z|<1.Letχ^b be the total service time of the first customer served in server busy period b, and J denotes the arrival number during χ^b. Since the length of b is irrelevant to the service order of customers and the customer arrival is a Bernoulli process with rate λ, we get
(3)P(J=j)=∑k=j∞‍g^kCkjλj(1-λ)k-j,j≥0.Denote byO1,O2,…,OJ the J arrivals during χ^b, called primary customers. Those who arrive after the primary customers are called secondary customers. Since the service order of customers has no influence on the length of b, we can apply the following service order: primary customers are served in the order of O1,O2,…,OJ. After serving each primary customer, the server will serve any secondary customer until there are no secondary customers present. Therefore we have the decomposition of b as follows:
(4)b=χ^b+l1+⋯+lJ,
where lk,1≤k≤J, denotes the time length from the epoch when the server begins to serve the kth primary customer until the next epoch when the service of the (k+1)th primary customer begins. Thus, lk(k=1,2,…,J) is mutually independent and has the same distribution as b. It is clear that l1+l2+⋯+lJ=0 if J=0. Noting that the point when the server busy period ends is a regenerative one, for j≥1, we get(5)Qj(n+)=P{χ^b+l1+⋯+lJ>n+,N(n+)=j}=P{χ^b>n+,N(n+)=j}+P{χ^b≤n+<χ^b+l1+⋯+lJ,N(n+)=j}=∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1+∑k=1n‍g^k∑i=0k‍Ckiλi(1-λ)k-i×P{l1+⋯+li>(n-k)+,N((n-k)+)=j}.By the above service order, if the epoch(n-k)+ is located in lm(1≤m≤i) and N((n-k)+)=j, then at epoch (n-k)+ there are i-m primary customers and j-(i-m) secondary customers in the system. From the fact that lm(1≤m≤i) obeys the same probability property as b and total probability formula, we obtain
(6)P{l1+⋯+li>(n-k)+,N((n-k)+)=j}=∑m=1i‍P{l1+⋯+lm-1≤(n-k)+<l1+⋯+lm,N((n-k)+)=j}=∑m=1i‍∑r=m-1n-k‍P{l1+⋯+lm-1=r}×P{lm>(n-k-r)+,N((n-k-r)+)=j-(i-m)}=∑m=1i‍∑r=m-1n-k‍P{l1+⋯+lm-1=r}×Qj-i+m((n-k-r)+).Substituting (6) into (5) and taking the PGF of (5), we get the expression of qj(z),j=1,2,…, as follows:
(7)qj(z)=1G^((1-λ)z)×{+b(z)∑n=j-1∞‍zn∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1}∑i=1j-1‍qj-i(z)bi(z)×[∑m=1i‍∑k=m∞‍b(z)-G^((1-λ)z)-∑m=1i‍∑k=m∞‍Ckmg^kzk(λb(z))m(1-λ)k-m]+b(z)∑n=j-1∞‍zn∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1},nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn|z|<1,
where ∑i=1j‍=0 if j≤0.


### 3.2. PGF for Conditional Distribution of Transient System Size at Any Epochn+

Denote byPij(n+)=P{N(n+)=j∣N(0+)=i}, i,j=0,1,2,… the conditional distribution of transient system size at epoch n+, and pij(z)=∑n=0∞‍Pij(n+)zn,|z|<1. Also, we denote αm=βm-1-βm, βm=P{∑j=1m‍χ^j<D}, m=1,2,…,D, β0=1, βD=0.LetIk and bk,k≥1 denote the kth system idle period and server busy period initiated with initial state N(0+)=0, respectively; then from the assumption that after the first busy period the server takes a modified D-policy, b1 obeys the same distribution as b (see Section 2.3). Noting that N(n+)=0 if and only if the epoch n+ is located in system idle period, we get
(8)P00(n+)=P{0≤n+<I1}+P{I1+b1≤n+<I1+b1+I2}+∑m=1D‍P{∑j=1m-1‍χ^j<D≤∑j=1m‍χ^j,I1+b1+I2+∑k=1m-1‍τk≤n+,N(n+)=0∑j=1m-1‍}.Since the customer arrivals are generated by Bernoulli process, the ending epoch of server busy periodb1 is a renewal point. So the second term of (8) is equal to
(9)∑u=2n‍P{I1+b1=u}(1-λ)n-u.The∑j=1m-1‍χ^j<D≤∑j=1m‍χ^j in the third term of (8) means that there are m arrivals at initiation point of the second busy period b2 of server. As the beginning epoch of b2 is a renewal point, using renewal process theory, the third term of (8) is expressed as
(10)∑m=1D‍αm∑u=2n-m‍P{I1+b1=u}×∑t=1n-u-m+1‍P{I2=t}∑r=m-1n-u-t‍P{∑k=1m-1‍τk=r}×Pm0((n-u-t-r)+).Multiplying (8) by zn and summing over n after substituting (9) and (10) into (8), it finally yields
(11)p00(z)=1-f(z)1-z+f(z)b(z)(1-f(z))1-z+b(z)∑m=1D‍αm[f(z)]m+1pm0(z),
where f(z)=λz/(1-(1-λ)z), and b(z) is given by Section 2.3.Fori≥1, Pi0(n+) shows that there are i customers at initial epoch 0+. Based on the supposition in Section 1, the server busy period b〈i〉 will begin at epoch 0+. Similar to the analysis of P00(n+), so we have
(12)Pi0(n+)=P{b〈i〉≤n+<b〈i〉+I1}+∑m=1D‍P{∑j=1m-1‍χ^j<D≤∑j=1m‍χ^j,b〈i〉+I1+∑k=1m-1‍τk≤n+,N(n+)=0∑j=1m-1‍}=∑u=in‍P{b〈i〉=u}(1-λ)n-u+∑m=1D‍αm∑u=in-m‍P{b〈i〉=u}×∑t=1n-u-m+1‍P{I1=t}×∑r=m-1n-u-t‍P{∑k=1m-1‍τk=r}×Pm0((n-u-t-r)+).
Taking the PGF on (12) gives
(13)pi0(z)=bi(z)(1-f(z))1-z+bi(z)∑m=1D‍αmfm(z)pm0(z),000000000000000000000i≥1.
Solving (11) and (13) yields
(14)p00(z)=1-f(z)1-z{1+f(z)b(z)1-∑m=1D‍αm[f(z)b(z)]m},0000000000000000000000000000000000000|z|<1,pi0(z)=[1-f(z)]bi(z)(1-z){1-∑m=1D‍αm[f(z)b(z)]m},00000000000000000000000000|z|<1,i≥1.For1≤j≤D-1, N(n+)=j represents the fact that the epoch n+ is in server busy period or server idle period with j customers being at present. By means of the same analysis method used for P00(n+), we have
(15)P0j(n+)=P{I1≤n+<I1+b1,N(n+)=j}+P{∑i=1j‍χ^i<D,I1+b1+I2+∑l=1j-1‍τl≤n+<I1+b1+I2+∑l=1j‍τl}+∑m=1D‍P{∑i=1m-1‍χ^i<D≤∑i=1m‍χ^i,I1+b1+I2+∑l=1m-1‍τl≤n+,N(n+)=j}.In the first term of (15), the expression of “I1≤n+<I1+b1, N(n+)=j” denotes the epoch n+ located in the first server busy period b1 and the customer number in system is j. From the fact that the beginning point of b1 is a renewal one, the first term of (15) is
(16)∑u=1n‍P{I1=u}Qj((n-u)+).In the second term of (15), the “∑i=1j‍χ^i<D, I1+b1+I2+∑l=1j-1‍τl≤n+<I1+b1+I2+∑l=1jτl” implies that the first arrival occurs during the second system idle period I2, the epoch n+ is located in the jth arrival-interval τj,j≥1 after I2, and at initiation point of b2 total arrival number exceeds j. It means that there are total j customers in the system at epoch n+ but the service does not begin, so the second term of (15) is given by
(17)βj∑u=j+2n‍P{I1+b1+I2+∑l=1j-1‍τl=u}P{τj>(n-u)+}.Being similar to the analysis of (10), the third term of (15) is given by
(18)∑m=1D‍αm∑u=m+2n‍P{I1+b1+I2+∑l=1m-1‍τl=u}Pmj((n-u)+).Multiplying (15) by zn and summing over n after substituting (16)–(18) into (15), we obtain, for j=1,2,…,D-1,
(19)p0j(z)=f(z)qj(z)+1-f(z)1-zfj+1(z)b(z)βj+b(z)∑m=1D‍αmfm+1(z)pmj(z).Fori≥1, similarly, we have
(20)Pij(n+)=P{b〈i〉>n+,N(n+)=j}+βj∑u=j+in‍P{b〈i〉+I1+∑l=1j-1‍τl=u}nnnnnnnnnnn×P{τj>(n-u)+}+∑m=1D‍αm∑u=m+in‍P{b〈i〉+I1+∑l=1m-1‍τl=u}nnnnnnnnnnnnnn×Pmj((n-u)+).Sinceb〈i〉 can be expressed by b〈i〉=ς1+⋯+ςi, where ς1,…,ςi are i.i.d. variables with the same distribution as the server busy period b, applying the decomposition technique used in (6), the first term in (20) is given by
(21)∑k=1i‍∑u=k-1n‍P{b〈k-1〉=u}Qj-i+k((n-u)+).Substituting (21) into (20), multiplying (20) by zn, and summing over n yield
(22)pij(z)=∑k=1i‍qj-i+k(z)bk-1(z)+1-f(z)1-zfj(z)bi(z)βj+bi(z)∑m=1D‍αmfm(z)pmj(z).Solving (19) and (22) leads to the expressions of p0j(z) and pij(z); that is, for |z|<1, i≥1, and j=1,2,…,D-1,
(23)p0j(z)=f(z)qj(z)+((1-f(z))/(1-z))fj+1(z)b(z)βj+f(z)φ(z)1-∑m=1D‍αm[f(z)b(z)]m,pij(z)=∑k=1i‍qj-i+k(z)bk-1(z)+((1-f(z))/(1-z))fj(z)bi(z)βj+bi-1(z)φ(z)1-∑m=1D‍αm[f(z)b(z)]m,
where φ(z)=∑m=1Dαmfm(z)∑k=1mbk(z)qj-m+k(z).Finally, sinceN(n+)=j, j≥D indicates that the epoch n+ is in server busy period, by similar operations in the analysis of pij(z), i≥0, j=1,2,…,D-1, we obtain, for |z|<1, i≥1, and j=D,D+1,D+2,…,
(24)p0j(z)=f(z)qj(z)+f(z)∑m=1Dαmfm(z)∑k=1mbk(z)qj-m+k(z)1-∑m=1Dαm[f(z)b(z)]m,pij(z)=∑k=1i‍qj-i+k(z)bk-1(z)+bi-1(z)∑m=1Dαmfm(z)∑k=1mbk(z)qj-m+k(z)1-∑m=1Dαm[f(z)b(z)]m.


### 3.3. Steady-State System Size Distribution at Any Epochn+

Now, based on the PGF of the transient-state system size distribution obtained above, we investigate the steady-state system size distribution at any epochn+.Letpj+=lim⁡n→∞⁡P{N(n+)=j},j≥0. According to the law of total probability and limitation theory of z-transform (see [24]), we get
(25)pj+=lim⁡n→∞∑i=0∞‍Pij(n+)P{N(0+)=i}=∑i=0∞‍P{N(0+)=i}lim⁡n→∞Pij(n+)=lim⁡z↑1(1-z)pij(z).
In (25), applying the expressions of pij(z),i,j≥0, L’Hospital’s rule, and (2) leads to(1)
forρ=λμ/θ≥1, pj+=0, j=0,1,2,…;(2)
forρ=λμ/θ<1, the steady-state system size distribution {pj+,j≥0} is given by
(26)p0+=1-ρ1+∑j=1D-1‍βj,pj+=p0+(βj+∑m=1D‍αm∑k=1m‍λqj-m+k(1)),nnnnnnnnnnnnnnj=1,2,…,D-1,pj+=p0+∑m=1D‍αm∑k=1m‍λqj-m+k(1),nnnnnn0nj=D,D+1,…,where
(27)q1(1)=1-G^(1-λ)λG^(1-λ),qj(1)=1G^(1-λ)×{∑n=j-1∞‍∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1nnn+∑i=1j-1‍qj-i(1)[∑m=1i‍1-G^(1-λ)nnnnnnnnnnnnnn-∑m=1i‍∑k=m∞‍g^kCkmλm(1-λ)k-m]},nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnj=2,3,….Remark 1.
Whenρ=λμ/θ<1, noting that ∑j=1∞‍∑k=1m‍qj-m+k(1)=mρ/(λ(1-ρ)) and ∑m=1D‍mαm=1+∑j=1D-1‍βj, we easily get ∑j=0∞‍pj+=1.
Letπ+(z)=∑j=0∞‍pj+zj, |z|<1 be the PGF of steady-state system size at epoch n+. For ρ=λμ/θ<1, calculating directly and noting
(28)∑j=1∞‍zj∑k=1m‍qj-m+k(1)=z(1-zm)[1-G^(1-λ+λz)]λ(1-z)[G^(1-λ+λz)-z],(1-z)(1+∑j=1D-1‍βjzj)=1-∑m=1D‍αmzm,
we have
(29)π+(z)=(1-ρ)(1-z)G^(1-λ+λz)G^(1-λ+λz)-z·1+∑j=1D-1‍βjzj1+∑j=1D-1‍βj.Remark 2.
It is observed from (29) that in our queue the steady-state system size Υ can be decomposed into the sum of two independent random variables: Υ=Υ0+Υ1. Υ0 is the steady-state system size of Geo/G/1 queue with Bernoulli feedback and has the PGF as ((1-ρ)(1-z)G^(1-λ+λz))/(G^(1-λ+λz)-z). Υ1 is the number of additional customers due to the modified D-policy and obeys the following probability distribution:
(30)P(Υ1=0)=11+∑j=1D-1‍βj,P(Υ1=j)=βj1+∑j=1D-1‍βj,nnnnnj=1,2,…,D-1.Remark 3.
It should be noted that the stochastic decomposition property of steady-state system size does not hold for the Geo/G/1 queue under the D-policy (see Lee et al. [16]).
Denote byE[L+] the expected steady-state system size at epoch n+; then by E[L+]=(dπ+(z)/dz)|z=1 and direct calculations, for ρ=λμ/θ<1, we can obtain
(31)E[L+]=ρ+λ2G^′′(1)2(1-ρ)+∑j=1D-1‍jβj1+∑j=1D-1‍βj,
where j≤0, ∑i=1j‍=0.


### 3.4. Steady-State System Size Distributions at Epochsn and n-

LetPij(n)=P{N(n)=j∣N(0)=i} and Pij(n-)=P{N(n-)=j∣N(0-)=i}, i,j=0,1,2,… be the conditional distributions of transient system size at epochs n and n-, pj=lim⁡n→∞⁡P{N(n)=j} and pj-=lim⁡n→∞⁡P{N(n-)=j}, j=0,1,2,… denote steady- state system size distributions at epochs n and n-, π(z)=∑j=0∞‍pjzj, π-(z)=∑j=0∞‍pj-zj, |z|<1, and E(L) is the expected steady-state system size at any epoch n. Also, it is assumed that no arrival occurs in the beginning time interval (0-,0) and no departure takes place in (0,0+).From the assumption that no arrival occurs in(0-,0) and no departure takes place in (0,0+), for i,j≥0,n≥1, we get P{N(0-)=j}=P{N(0)=j}=P{N(0+)=j}. So
(32)Pij(n-)=P{N(n-)=j∣N(0-)=i}=P{N((n-1)+)=j∣N(0+)=i}=Pij((n-1)+).
Taking limit as n→∞ under the stability condition, ρ=λμ/θ<1, we get
(33)pj-=pj+,j≥0,which  is  equivalent  toπ-(z)=π+(z),|z|<1.As the customers arrive in the system according to Bernoulli process, and the departure occurs only in interval(n,n+), for ρ=λμ/θ<1, we obtain p0=(1-λ)p0-, pj=(1-λ)pj-+λpj-1-, j=1,2,…. By using (33), π+(z) in (29), and E(L+) in (31), when ρ=λμ/θ<1, we have
(34)p0=(1-λ)p0+,pj=(1-λ)pj++λpj-1+,nnnnnnnnnnnnnnnnnnnnnnnnj=1,2,…,(35)π(z)=(1-λ+λz)(1-ρ)(1-z)G^(1-λ+λz)G^(1-λ+λz)-z·1+∑j=1D-1‍βjzj1+∑j=1D-1‍βj,(36)E(L)=λ+ρ+λ2G^′′(1)2(1-ρ)+∑j=1D-1‍jβj1+∑j=1D-1‍βj.Remark 4.
At epochsn-, n+, and n, the following relations are clear:(1)
pj-=pj+≠pj, j=0,1,2,…, or π-(z)=π+(z)≠π(z);(2)
p0-=p0+>p0, E(L-)=E(L+)<E(L).


## 3.1. PGF for Transient System Size Distribution at Epochn+ in Server Busy Period b

Denote byQj(n+)=P{b>n+,N(n+)=j},j≥1 transient system size distribution at epoch n+ in server busy period b, and its PGF is qj(z)=∑n=0∞‍Qj(n+)zn, |z|<1.Letχ^b be the total service time of the first customer served in server busy period b, and J denotes the arrival number during χ^b. Since the length of b is irrelevant to the service order of customers and the customer arrival is a Bernoulli process with rate λ, we get
(3)P(J=j)=∑k=j∞‍g^kCkjλj(1-λ)k-j,j≥0.Denote byO1,O2,…,OJ the J arrivals during χ^b, called primary customers. Those who arrive after the primary customers are called secondary customers. Since the service order of customers has no influence on the length of b, we can apply the following service order: primary customers are served in the order of O1,O2,…,OJ. After serving each primary customer, the server will serve any secondary customer until there are no secondary customers present. Therefore we have the decomposition of b as follows:
(4)b=χ^b+l1+⋯+lJ,
where lk,1≤k≤J, denotes the time length from the epoch when the server begins to serve the kth primary customer until the next epoch when the service of the (k+1)th primary customer begins. Thus, lk(k=1,2,…,J) is mutually independent and has the same distribution as b. It is clear that l1+l2+⋯+lJ=0 if J=0. Noting that the point when the server busy period ends is a regenerative one, for j≥1, we get(5)Qj(n+)=P{χ^b+l1+⋯+lJ>n+,N(n+)=j}=P{χ^b>n+,N(n+)=j}+P{χ^b≤n+<χ^b+l1+⋯+lJ,N(n+)=j}=∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1+∑k=1n‍g^k∑i=0k‍Ckiλi(1-λ)k-i×P{l1+⋯+li>(n-k)+,N((n-k)+)=j}.By the above service order, if the epoch(n-k)+ is located in lm(1≤m≤i) and N((n-k)+)=j, then at epoch (n-k)+ there are i-m primary customers and j-(i-m) secondary customers in the system. From the fact that lm(1≤m≤i) obeys the same probability property as b and total probability formula, we obtain
(6)P{l1+⋯+li>(n-k)+,N((n-k)+)=j}=∑m=1i‍P{l1+⋯+lm-1≤(n-k)+<l1+⋯+lm,N((n-k)+)=j}=∑m=1i‍∑r=m-1n-k‍P{l1+⋯+lm-1=r}×P{lm>(n-k-r)+,N((n-k-r)+)=j-(i-m)}=∑m=1i‍∑r=m-1n-k‍P{l1+⋯+lm-1=r}×Qj-i+m((n-k-r)+).Substituting (6) into (5) and taking the PGF of (5), we get the expression of qj(z),j=1,2,…, as follows:
(7)qj(z)=1G^((1-λ)z)×{+b(z)∑n=j-1∞‍zn∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1}∑i=1j-1‍qj-i(z)bi(z)×[∑m=1i‍∑k=m∞‍b(z)-G^((1-λ)z)-∑m=1i‍∑k=m∞‍Ckmg^kzk(λb(z))m(1-λ)k-m]+b(z)∑n=j-1∞‍zn∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1},nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn|z|<1,
where ∑i=1j‍=0 if j≤0.


## 3.2. PGF for Conditional Distribution of Transient System Size at Any Epochn+

Denote byPij(n+)=P{N(n+)=j∣N(0+)=i}, i,j=0,1,2,… the conditional distribution of transient system size at epoch n+, and pij(z)=∑n=0∞‍Pij(n+)zn,|z|<1. Also, we denote αm=βm-1-βm, βm=P{∑j=1m‍χ^j<D}, m=1,2,…,D, β0=1, βD=0.LetIk and bk,k≥1 denote the kth system idle period and server busy period initiated with initial state N(0+)=0, respectively; then from the assumption that after the first busy period the server takes a modified D-policy, b1 obeys the same distribution as b (see Section 2.3). Noting that N(n+)=0 if and only if the epoch n+ is located in system idle period, we get
(8)P00(n+)=P{0≤n+<I1}+P{I1+b1≤n+<I1+b1+I2}+∑m=1D‍P{∑j=1m-1‍χ^j<D≤∑j=1m‍χ^j,I1+b1+I2+∑k=1m-1‍τk≤n+,N(n+)=0∑j=1m-1‍}.Since the customer arrivals are generated by Bernoulli process, the ending epoch of server busy periodb1 is a renewal point. So the second term of (8) is equal to
(9)∑u=2n‍P{I1+b1=u}(1-λ)n-u.The∑j=1m-1‍χ^j<D≤∑j=1m‍χ^j in the third term of (8) means that there are m arrivals at initiation point of the second busy period b2 of server. As the beginning epoch of b2 is a renewal point, using renewal process theory, the third term of (8) is expressed as
(10)∑m=1D‍αm∑u=2n-m‍P{I1+b1=u}×∑t=1n-u-m+1‍P{I2=t}∑r=m-1n-u-t‍P{∑k=1m-1‍τk=r}×Pm0((n-u-t-r)+).Multiplying (8) by zn and summing over n after substituting (9) and (10) into (8), it finally yields
(11)p00(z)=1-f(z)1-z+f(z)b(z)(1-f(z))1-z+b(z)∑m=1D‍αm[f(z)]m+1pm0(z),
where f(z)=λz/(1-(1-λ)z), and b(z) is given by Section 2.3.Fori≥1, Pi0(n+) shows that there are i customers at initial epoch 0+. Based on the supposition in Section 1, the server busy period b〈i〉 will begin at epoch 0+. Similar to the analysis of P00(n+), so we have
(12)Pi0(n+)=P{b〈i〉≤n+<b〈i〉+I1}+∑m=1D‍P{∑j=1m-1‍χ^j<D≤∑j=1m‍χ^j,b〈i〉+I1+∑k=1m-1‍τk≤n+,N(n+)=0∑j=1m-1‍}=∑u=in‍P{b〈i〉=u}(1-λ)n-u+∑m=1D‍αm∑u=in-m‍P{b〈i〉=u}×∑t=1n-u-m+1‍P{I1=t}×∑r=m-1n-u-t‍P{∑k=1m-1‍τk=r}×Pm0((n-u-t-r)+).
Taking the PGF on (12) gives
(13)pi0(z)=bi(z)(1-f(z))1-z+bi(z)∑m=1D‍αmfm(z)pm0(z),000000000000000000000i≥1.
Solving (11) and (13) yields
(14)p00(z)=1-f(z)1-z{1+f(z)b(z)1-∑m=1D‍αm[f(z)b(z)]m},0000000000000000000000000000000000000|z|<1,pi0(z)=[1-f(z)]bi(z)(1-z){1-∑m=1D‍αm[f(z)b(z)]m},00000000000000000000000000|z|<1,i≥1.For1≤j≤D-1, N(n+)=j represents the fact that the epoch n+ is in server busy period or server idle period with j customers being at present. By means of the same analysis method used for P00(n+), we have
(15)P0j(n+)=P{I1≤n+<I1+b1,N(n+)=j}+P{∑i=1j‍χ^i<D,I1+b1+I2+∑l=1j-1‍τl≤n+<I1+b1+I2+∑l=1j‍τl}+∑m=1D‍P{∑i=1m-1‍χ^i<D≤∑i=1m‍χ^i,I1+b1+I2+∑l=1m-1‍τl≤n+,N(n+)=j}.In the first term of (15), the expression of “I1≤n+<I1+b1, N(n+)=j” denotes the epoch n+ located in the first server busy period b1 and the customer number in system is j. From the fact that the beginning point of b1 is a renewal one, the first term of (15) is
(16)∑u=1n‍P{I1=u}Qj((n-u)+).In the second term of (15), the “∑i=1j‍χ^i<D, I1+b1+I2+∑l=1j-1‍τl≤n+<I1+b1+I2+∑l=1jτl” implies that the first arrival occurs during the second system idle period I2, the epoch n+ is located in the jth arrival-interval τj,j≥1 after I2, and at initiation point of b2 total arrival number exceeds j. It means that there are total j customers in the system at epoch n+ but the service does not begin, so the second term of (15) is given by
(17)βj∑u=j+2n‍P{I1+b1+I2+∑l=1j-1‍τl=u}P{τj>(n-u)+}.Being similar to the analysis of (10), the third term of (15) is given by
(18)∑m=1D‍αm∑u=m+2n‍P{I1+b1+I2+∑l=1m-1‍τl=u}Pmj((n-u)+).Multiplying (15) by zn and summing over n after substituting (16)–(18) into (15), we obtain, for j=1,2,…,D-1,
(19)p0j(z)=f(z)qj(z)+1-f(z)1-zfj+1(z)b(z)βj+b(z)∑m=1D‍αmfm+1(z)pmj(z).Fori≥1, similarly, we have
(20)Pij(n+)=P{b〈i〉>n+,N(n+)=j}+βj∑u=j+in‍P{b〈i〉+I1+∑l=1j-1‍τl=u}nnnnnnnnnnn×P{τj>(n-u)+}+∑m=1D‍αm∑u=m+in‍P{b〈i〉+I1+∑l=1m-1‍τl=u}nnnnnnnnnnnnnn×Pmj((n-u)+).Sinceb〈i〉 can be expressed by b〈i〉=ς1+⋯+ςi, where ς1,…,ςi are i.i.d. variables with the same distribution as the server busy period b, applying the decomposition technique used in (6), the first term in (20) is given by
(21)∑k=1i‍∑u=k-1n‍P{b〈k-1〉=u}Qj-i+k((n-u)+).Substituting (21) into (20), multiplying (20) by zn, and summing over n yield
(22)pij(z)=∑k=1i‍qj-i+k(z)bk-1(z)+1-f(z)1-zfj(z)bi(z)βj+bi(z)∑m=1D‍αmfm(z)pmj(z).Solving (19) and (22) leads to the expressions of p0j(z) and pij(z); that is, for |z|<1, i≥1, and j=1,2,…,D-1,
(23)p0j(z)=f(z)qj(z)+((1-f(z))/(1-z))fj+1(z)b(z)βj+f(z)φ(z)1-∑m=1D‍αm[f(z)b(z)]m,pij(z)=∑k=1i‍qj-i+k(z)bk-1(z)+((1-f(z))/(1-z))fj(z)bi(z)βj+bi-1(z)φ(z)1-∑m=1D‍αm[f(z)b(z)]m,
where φ(z)=∑m=1Dαmfm(z)∑k=1mbk(z)qj-m+k(z).Finally, sinceN(n+)=j, j≥D indicates that the epoch n+ is in server busy period, by similar operations in the analysis of pij(z), i≥0, j=1,2,…,D-1, we obtain, for |z|<1, i≥1, and j=D,D+1,D+2,…,
(24)p0j(z)=f(z)qj(z)+f(z)∑m=1Dαmfm(z)∑k=1mbk(z)qj-m+k(z)1-∑m=1Dαm[f(z)b(z)]m,pij(z)=∑k=1i‍qj-i+k(z)bk-1(z)+bi-1(z)∑m=1Dαmfm(z)∑k=1mbk(z)qj-m+k(z)1-∑m=1Dαm[f(z)b(z)]m.


## 3.3. Steady-State System Size Distribution at Any Epochn+

Now, based on the PGF of the transient-state system size distribution obtained above, we investigate the steady-state system size distribution at any epochn+.Letpj+=lim⁡n→∞⁡P{N(n+)=j},j≥0. According to the law of total probability and limitation theory of z-transform (see [24]), we get
(25)pj+=lim⁡n→∞∑i=0∞‍Pij(n+)P{N(0+)=i}=∑i=0∞‍P{N(0+)=i}lim⁡n→∞Pij(n+)=lim⁡z↑1(1-z)pij(z).
In (25), applying the expressions of pij(z),i,j≥0, L’Hospital’s rule, and (2) leads to(1)
forρ=λμ/θ≥1, pj+=0, j=0,1,2,…;(2)
forρ=λμ/θ<1, the steady-state system size distribution {pj+,j≥0} is given by
(26)p0+=1-ρ1+∑j=1D-1‍βj,pj+=p0+(βj+∑m=1D‍αm∑k=1m‍λqj-m+k(1)),nnnnnnnnnnnnnnj=1,2,…,D-1,pj+=p0+∑m=1D‍αm∑k=1m‍λqj-m+k(1),nnnnnn0nj=D,D+1,…,where
(27)q1(1)=1-G^(1-λ)λG^(1-λ),qj(1)=1G^(1-λ)×{∑n=j-1∞‍∑k=n+1∞‍g^kCnj-1λj-1(1-λ)n-j+1nnn+∑i=1j-1‍qj-i(1)[∑m=1i‍1-G^(1-λ)nnnnnnnnnnnnnn-∑m=1i‍∑k=m∞‍g^kCkmλm(1-λ)k-m]},nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnj=2,3,….Remark 1.
Whenρ=λμ/θ<1, noting that ∑j=1∞‍∑k=1m‍qj-m+k(1)=mρ/(λ(1-ρ)) and ∑m=1D‍mαm=1+∑j=1D-1‍βj, we easily get ∑j=0∞‍pj+=1.
Letπ+(z)=∑j=0∞‍pj+zj, |z|<1 be the PGF of steady-state system size at epoch n+. For ρ=λμ/θ<1, calculating directly and noting
(28)∑j=1∞‍zj∑k=1m‍qj-m+k(1)=z(1-zm)[1-G^(1-λ+λz)]λ(1-z)[G^(1-λ+λz)-z],(1-z)(1+∑j=1D-1‍βjzj)=1-∑m=1D‍αmzm,
we have
(29)π+(z)=(1-ρ)(1-z)G^(1-λ+λz)G^(1-λ+λz)-z·1+∑j=1D-1‍βjzj1+∑j=1D-1‍βj.Remark 2.
It is observed from (29) that in our queue the steady-state system size Υ can be decomposed into the sum of two independent random variables: Υ=Υ0+Υ1. Υ0 is the steady-state system size of Geo/G/1 queue with Bernoulli feedback and has the PGF as ((1-ρ)(1-z)G^(1-λ+λz))/(G^(1-λ+λz)-z). Υ1 is the number of additional customers due to the modified D-policy and obeys the following probability distribution:
(30)P(Υ1=0)=11+∑j=1D-1‍βj,P(Υ1=j)=βj1+∑j=1D-1‍βj,nnnnnj=1,2,…,D-1.Remark 3.
It should be noted that the stochastic decomposition property of steady-state system size does not hold for the Geo/G/1 queue under the D-policy (see Lee et al. [16]).
Denote byE[L+] the expected steady-state system size at epoch n+; then by E[L+]=(dπ+(z)/dz)|z=1 and direct calculations, for ρ=λμ/θ<1, we can obtain
(31)E[L+]=ρ+λ2G^′′(1)2(1-ρ)+∑j=1D-1‍jβj1+∑j=1D-1‍βj,
where j≤0, ∑i=1j‍=0.


## 3.4. Steady-State System Size Distributions at Epochsn and n-

LetPij(n)=P{N(n)=j∣N(0)=i} and Pij(n-)=P{N(n-)=j∣N(0-)=i}, i,j=0,1,2,… be the conditional distributions of transient system size at epochs n and n-, pj=lim⁡n→∞⁡P{N(n)=j} and pj-=lim⁡n→∞⁡P{N(n-)=j}, j=0,1,2,… denote steady- state system size distributions at epochs n and n-, π(z)=∑j=0∞‍pjzj, π-(z)=∑j=0∞‍pj-zj, |z|<1, and E(L) is the expected steady-state system size at any epoch n. Also, it is assumed that no arrival occurs in the beginning time interval (0-,0) and no departure takes place in (0,0+).From the assumption that no arrival occurs in(0-,0) and no departure takes place in (0,0+), for i,j≥0,n≥1, we get P{N(0-)=j}=P{N(0)=j}=P{N(0+)=j}. So
(32)Pij(n-)=P{N(n-)=j∣N(0-)=i}=P{N((n-1)+)=j∣N(0+)=i}=Pij((n-1)+).
Taking limit as n→∞ under the stability condition, ρ=λμ/θ<1, we get
(33)pj-=pj+,j≥0,which  is  equivalent  toπ-(z)=π+(z),|z|<1.As the customers arrive in the system according to Bernoulli process, and the departure occurs only in interval(n,n+), for ρ=λμ/θ<1, we obtain p0=(1-λ)p0-, pj=(1-λ)pj-+λpj-1-, j=1,2,…. By using (33), π+(z) in (29), and E(L+) in (31), when ρ=λμ/θ<1, we have
(34)p0=(1-λ)p0+,pj=(1-λ)pj++λpj-1+,nnnnnnnnnnnnnnnnnnnnnnnnj=1,2,…,(35)π(z)=(1-λ+λz)(1-ρ)(1-z)G^(1-λ+λz)G^(1-λ+λz)-z·1+∑j=1D-1‍βjzj1+∑j=1D-1‍βj,(36)E(L)=λ+ρ+λ2G^′′(1)2(1-ρ)+∑j=1D-1‍jβj1+∑j=1D-1‍βj.Remark 4.
At epochsn-, n+, and n, the following relations are clear:(1)
pj-=pj+≠pj, j=0,1,2,…, or π-(z)=π+(z)≠π(z);(2)
p0-=p0+>p0, E(L-)=E(L+)<E(L).


## 4. Special Cases

Case 1.
Letθ=1 and D=1; then our model becomes a discrete-time Geo/G/1 queue with LAS-DA. In this case, for ρ=λμ<1, we have
(37)p0+=1-ρ,pj+=λ(1-ρ)qj(1),j=1,2,…,(38)π+(z)=(1-ρ)(1-z)G(1-λ+λz)G(1-λ+λz)-z,(39)E[L+]=ρ+λ22(1-ρ)G′′(1),
where qj(1),j=1,2,…, are determined by (7), in which g^j=gj and G^(z)=G(z).Case 2.
Letθ=1, D=1, and the service times χi,i≥1 are independent and follow an identical geometric distribution with parameter δ(>0). In the present queue with early arrival system (EAS) policy, the steady-state system size distribution at any epoch n is given by
(40)pj(e)=1-ρ1-λ(ρ-λ1-λ)j,j=0,1,…,ifρ=λδ<1,
which agrees with that in [24].In fact, denote byN(e)(n) and N(e)(n+) the system sizes at epochs n and n+ in the present queue with early arrival system (EAS) policy, and {pj(e)} and {pj(e)+} are the steady-state system size distributions at any epochs n and n+. By the relation of customer number between late arrival system with delay access (LAS-DA) and early arrival system (EAS), we get N(n+)=N(e)(n+). So pj+=pj(e)+,j≥0. Since the customer arrival is a Bernoulli process with rate λ, we have
(41)p0+=p0(e)+=(1-λ)p0(e),pj+=pj(e)+=(1-λ)pj(e)+λpj-1(e),j≥1.
It follows from (41) that, for ρ=λ/δ<1 and |z|<1, the relation between π+(z)=∑j=0∞‍pj+zj and π(e)(z)=∑j=0∞‍pj(e)zj is
(42)π+(z)=(1-λ+λz)π(e)(z).
Using π+(z) in (38) and noting that the service times obey geometric distribution with mean 1/δ, we obtain
(43)π+(z)=(1-ρ)(1-λ+λz)1-λ+λz-ρz.
From (42) and (43), we get
(44)∑j=0∞‍pj(e)zj=π(e)(z)=1-ρ1-λ+λz-ρz=1-ρ1-λ∑j=0∞‍(ρ-λ1-λ)jzj.
This gives rise to (40).


## 5. Numerical Examples

One example of an application fitting our model is the wireless local area network (WLAN) introduced in Section1. In such a system, because the receptions and transmissions of radio signals are proceeded in fixed time intervals, the arrival of radio signals can be modelled as a Bernoulli process with rate λ, and the transmission time of each radio signal follows a discrete-time distribution (for numerical analysis, we assume that the transmission time is geometrically distributed with mean μ). Because of power saving, the AP is designed to start serving exhaustively whenever the workload of radio signals reaches a predetermined positive integer D. If a radio signal is transmitted with errors at the destination, it is retransmitted (feedback) with probability 1-θ.To illustrate the applicability and important application in system capacity design of steady-state system size distributions derived in Section3, based on the results obtained in (26) and (34), we numerically get the steady-state radio signal number distributions {pj,j≥0} and {pj+,j≥0} for the set of parameters (λ,μ,θ,D)=(0.25,1,0.4,20) (see Figure 1). Also, by using (31) and (36), the mean radio signal numbers E(L+) and E(L) and the intensity ρ for the above WLAN are obtained (see Table 1).Table 1
Steady-state radio signal number distributions at epochsn and n+ for a wireless local area network (λ=0.25, μ=1, θ=0.4, and D=20).
j
p
j
p
j
+
P
{
L
+
>
j
}
j
p
j
p
j
+
P
{
L
+
>
j
}
0
0.03270
0.04360
1.00000
13
0.00907
0.00763
0.09753
1
0.07085
0.07993
0.95639
14
0.00532
0.00455
0.08989
2
0.09327
0.09772
0.87645
15
0.00340
0.00301
0.08534
3
0.10115
0.10229
0.77873
16
0.00240
0.00220
0.08232
4
0.10264
0.10276
0.67643
17
0.00182
0.00169
0.08012
5
0.10115
0.10061
0.57367
18
0.00139
0.00129
0.07842
6
0.09636
0.09495
0.47305
19
0.00104
0.00096
0.07713
7
0.08727
0.08471
0.37810
20
0.00075
0.00068
0.07616
8
0.07375
0.07009
0.29338
21
0.00051
0.00046
0.07548
9
0.05728
0.05301
0.22329
22
0.00033
0.00029
0.07502
10
0.04058
0.03643
0.17027
23
0.00021
0.00018
0.07472
11
0.02628
0.02290
0.133836
24
0.00013
0.00011
0.07453
12
0.01577
0.01339
0.11093
25
0.00008
0.00006
0.07442
∑
j
=
0
25
p
j
=
0.92563, ∑j=025pj+=0.92564, E(L)=5.56511, E(L+)=5.31511, and ρ=0.33333<1.Figure 1
Steady-state radio signal number distributions at epochsn and n+ for a wireless local area network.Remark 5.
With the software of Matlab 7.0, the data in Table1 are accurate to five decimal places.
From the two curves of Figure1, we conclude that the steady-state distribution probabilities {pj,j≥0} and {pj+,j≥0} increase rapidly at an early stage and then decrease as radio signal number becomes large. Moreover, the decrease becomes stable when radio signal number is greater than 15. It is clear that the values of pj and pj+ have a small difference. Since pj+ or pj,j>20 tends to zero, it is improper for this WLAN to design an enough large space capacity.
LetK denote network space capacity, and L+ is radio signal number of WLAN at epoch n+; then from Table 1 we see that the overflow probability of WLAN P(L+>K) is smaller for larger network space capacity K, which is as expected. Also, from the data in Table 1, we get P{L+>E(L+)}=0.57367. Therefore, it is inappropriate to determine network space capacity with E(L+) because the overflow probability of WLAN is very large.


## 6. Conclusions

This paper studies the steady-state system size distribution for a discrete-time modified D-policy Geo/G/1 queue with feedback. The steady-state system size distributions at three-time epochs are obtained. It should be noted that the steady-state system size distributions presented by this paper are some recursive formulae, which are applicable to accurate numerical calculations. In the future, the system size distribution study of more complex discrete-time queues, such as the modified D-policyGeoX/G/1 queue with feedback and vacation, will be investigated by the similar analysis techniques.



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*Source: 345129-2014-06-22.xml*