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

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 epoch 𝑛⁺ 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.

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 which 𝐷 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 Section 3 is very useful in network capacity design (see Section 5).

This paper is organized as follows. The next section presents assumptions and preliminaries. Section 3 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, {𝜏_𝑖,𝑖≥1}, are independent identically distributed (i.i.d) random variables following a geometric distribution 𝑃{𝜏_𝑖=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 {𝜒_𝑛,𝑛≥1}, are i.i.d random variables obeying an arbitrary distribution 𝑔_𝑗=𝑃{𝜒_𝑛=𝑗},𝑗≥1, with probability generating function (PGF) 𝐺(𝑧)=∑^∞_𝑗=1𝑔_𝑗𝑧^𝑗, |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

Let 𝜒^𝑖_𝑛, 𝜒_𝑛, and Ω_𝑛 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(Ω_𝑛=𝑘)=𝜃(1−𝜃)^𝑘−1,𝑘≥1, and 𝜒_𝑛=∑^Ω𝑛_𝑖=1𝜒^𝑖_𝑛,𝑛≥1 are i.i.d. random variables, where 𝜒^𝑖_𝑛,𝑖,𝑛≥1 are mutually independent and have the same distribution as 𝜒_𝑛,𝑛≥1.

Further, the probability distribution 𝑔_𝑗, PGF 𝐺(𝑧), and mean 𝐸(𝜒) of total service time 𝜒 for one customer before departure are given, respectively, by 𝑔_𝑗=𝑃𝜒=𝑗=𝑃^Ω𝑛_𝑖=1𝜒^𝑖_𝑛==𝑗_𝑗𝑘=1𝜃(1−𝜃)^𝑘−1𝑃^𝑘_𝑖=1𝜒^𝑖_𝑛,𝐺(𝑧)==𝑗𝑗≥1,_∞𝑗=1𝑔_𝑗𝑧^𝑗=𝜃𝐺(𝑧)()𝐺(𝑧),|𝑧|𝐸𝜒=d𝐺(𝑧)1−1−𝜃<1,d𝑧_𝑧=1=𝜇𝜃. (1)

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.

Let 𝐼_𝑖 denote the 𝑖th system idle period initiated with the initial state 𝑁(0⁺)(≥0); then it follows from the queue assumptions that {𝐼_𝑖,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 by 𝑏 the server busy period initiated with one customer and its PGF is given by 𝑏(𝑧)=∑^∞_𝑗=0𝑃(𝑏=𝑗)𝑧^𝑗, |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 𝐸(𝑏)=𝜇𝜃−𝜆𝜇,𝜌<1,∞,𝜌≥1, (2) where 𝜌=𝜆𝜇/𝜃 represents the traffic intensity of the considered queue.

Let 𝑏^⟨𝑖⟩ denote the server busy period initiated with 𝑖 customers; then 𝑏^⟨𝑖⟩ can be expressed as 𝑏^⟨𝑖⟩=𝜍₁+⋯+𝜍_𝑖, where 𝜍₁,…,𝜍_𝑖 are mutually independent with the same distribution as 𝑏. So the PGF of 𝑏^⟨𝑖⟩ is given by 𝑏^𝑖(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 epoch 𝑛⁺. Lastly we derive the steady-state system size distributions at epochs 𝑛⁻ and 𝑛.

3.1. PGF for Transient System Size Distribution at Epoch 𝑛⁺ in Server Busy Period 𝑏

Denote by 𝑄_𝑗(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 𝑞_𝑗(𝑧)=∑^∞_𝑛=0𝑄_𝑗(𝑛⁺)𝑧^𝑛, |𝑧|<1.

Let 𝜒_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 𝑃𝐽=𝑗=^∞_𝑘=𝑗𝑔_𝑘𝐶^𝑗_𝑘𝜆^𝑗(1−𝜆)^𝑘−𝑗,𝑗≥0. (3)

Denote by 𝑂₁,𝑂₂,…,𝑂_𝐽 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,…,𝑂_𝐽. 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: 𝑏=𝜒_𝑏+𝑙₁+⋯+𝑙_𝐽, (4) where 𝑙_𝑘,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, 𝑙_𝑘(𝑘=1,2,…,𝐽) is mutually independent and has the same distribution as 𝑏. It is clear that 𝑙₁+𝑙₂+⋯+𝑙_𝐽=0 if 𝐽=0. Noting that the point when the server busy period ends is a regenerative one, for 𝑗≥1, we get𝑄_𝑗𝑛⁺=𝑃𝜒_𝑏+𝑙₁+⋯+𝑙_𝐽>𝑛⁺,𝑁𝑛⁺𝜒=𝑗=𝑃_𝑏>𝑛⁺,𝑁𝑛⁺𝜒=𝑗+𝑃_𝑏≤𝑛⁺<𝜒_𝑏+𝑙₁+⋯+𝑙_𝐽,𝑁𝑛⁺==𝑗_{∞𝑘=𝑛+1}𝑔_𝑘𝐶_𝑛^𝑗−1𝜆^𝑗−1(1−𝜆)^{𝑛−𝑗+1}+^𝑛_𝑘=1𝑔_𝑘^𝑘_𝑖=0𝐶^𝑖_𝑘𝜆^𝑖(1−𝜆)^𝑘−𝑖×𝑃𝑙₁+⋯+𝑙_𝑖>(𝑛−𝑘)⁺,𝑁(𝑛−𝑘)⁺=𝑗. (5)

By the above service order, if the epoch (𝑛−𝑘)⁺ is located in 𝑙_𝑚(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 𝑙_𝑚(1≤𝑚≤𝑖) obeys the same probability property as 𝑏 and total probability formula, we obtain 𝑃𝑙₁+⋯+𝑙_𝑖>(𝑛−𝑘)⁺,𝑁(𝑛−𝑘)⁺==𝑗_𝑖𝑚=1𝑃𝑙₁+⋯+𝑙_𝑚−1≤(𝑛−𝑘)⁺<𝑙₁+⋯+𝑙_𝑚,𝑁(𝑛−𝑘)⁺==𝑗_{𝑖𝑚=1𝑛−𝑘𝑟=𝑚−1}𝑃𝑙₁+⋯+𝑙_𝑚−1𝑙=𝑟×𝑃_𝑚>(𝑛−𝑘−𝑟)⁺,𝑁(𝑛−𝑘−𝑟)⁺(==𝑗−𝑖−𝑚)_{𝑖𝑚=1𝑛−𝑘𝑟=𝑚−1}𝑃𝑙₁+⋯+𝑙_𝑚−1=𝑟×𝑄_{𝑗−𝑖+𝑚}(𝑛−𝑘−𝑟)⁺. (6)

Substituting (6) into (5) and taking the PGF of (5), we get the expression of 𝑞_𝑗(𝑧),𝑗=1,2,…, as follows: 𝑞_𝑗(𝑧)=1𝐺)𝑧)×((1−𝜆_{𝑗−1𝑖=1}𝑞_𝑗−𝑖(𝑧)𝑏^𝑖(𝑧)×𝑏(𝑧)−𝐺)𝑧)−((1−𝜆_{𝑖∞𝑚=1𝑘=𝑚}𝐶^𝑚_𝑘𝑔_𝑘𝑧^𝑘(𝜆𝑏(𝑧))^𝑚(1−𝜆)^𝑘−𝑚+𝑏(𝑧)^∞_{𝑛=𝑗−1}𝑧^𝑛∞_𝑘=𝑛+1𝑔_𝑘𝐶_𝑛^𝑗−1𝜆^𝑗−1(1−𝜆)^{𝑛−𝑗+1},|𝑧|<1, (7) where ∑^𝑗_𝑖=1=0 if 𝑗≤0.

3.2. PGF for Conditional Distribution of Transient System Size at Any Epoch 𝑛⁺

Denote by 𝑃_𝑖𝑗(𝑛⁺)=𝑃{𝑁(𝑛⁺)=𝑗∣𝑁(0⁺)=𝑖}, 𝑖,𝑗=0,1,2,… the conditional distribution of transient system size at epoch 𝑛⁺, and 𝑝_𝑖𝑗(𝑧)=∑^∞_𝑛=0𝑃_𝑖𝑗(𝑛⁺)𝑧^𝑛,|𝑧|<1. Also, we denote 𝛼_𝑚=𝛽_𝑚−1−𝛽_𝑚, 𝛽_𝑚=𝑃{∑^𝑚_𝑗=1𝜒_𝑗<𝐷}, 𝑚=1,2,…,𝐷, 𝛽₀=1, 𝛽_𝐷=0.

Let 𝐼_𝑘 and 𝑏_𝑘,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 𝑃₀₀𝑛⁺=𝑃0≤𝑛⁺<𝐼₁+𝑃𝐼₁+𝑏₁≤𝑛⁺<𝐼₁+𝑏₁+𝐼₂+^𝐷_𝑚=1𝑃^𝑚−1_𝑗=1𝜒_𝑗<𝐷≤^𝑚_𝑗=1𝜒_𝑗,𝐼₁+𝑏₁+𝐼₂+^𝑚−1_𝑘=1𝜏_𝑘≤𝑛⁺,𝑁𝑛⁺=0. (8)

Since the customer arrivals are generated by Bernoulli process, the ending epoch of server busy period 𝑏₁ is a renewal point. So the second term of (8) is equal to ^𝑛_𝑢=2𝑃𝐼₁+𝑏₁=𝑢(1−𝜆)^𝑛−𝑢. (9)

The ∑^𝑚−1_𝑗=1𝜒_𝑗<𝐷≤∑^𝑚_𝑗=1𝜒_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 ^𝐷_𝑚=1𝛼_𝑚^𝑛−𝑚_𝑢=2𝑃𝐼₁+𝑏₁×=𝑢_{𝑛−𝑢−𝑚+1𝑡=1}𝑃𝐼₂=𝑡^{𝑛−𝑢−𝑡}_{𝑟=𝑚−1}𝑃^𝑚−1_𝑘=1𝜏_𝑘=𝑟×𝑃_𝑚0(𝑛−𝑢−𝑡−𝑟)⁺. (10)

Multiplying (8) by 𝑧^𝑛 and summing over 𝑛 after substituting (9) and (10) into (8), it finally yields 𝑝₀₀(𝑧)=1−𝑓(𝑧)+𝑓(𝑧)𝑏(𝑧)(𝑧)1−𝑧1−𝑓(𝑧)1−𝑧+𝑏^𝐷_𝑚=1𝛼_𝑚𝑓(𝑧)^𝑚+1𝑝_𝑚0(𝑧), (11) where 𝑓(𝑧)=𝜆𝑧/(1−(1−𝜆)𝑧), and 𝑏(𝑧) is given by Section 2.3.

For 𝑖≥1, 𝑃_𝑖0(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 𝑃_𝑖0𝑛⁺=𝑃𝑏^⟨𝑖⟩≤𝑛⁺<𝑏^⟨𝑖⟩+𝐼₁+^𝐷_𝑚=1𝑃^𝑚−1_𝑗=1𝜒_𝑗<𝐷≤^𝑚_𝑗=1𝜒_𝑗,𝑏^⟨𝑖⟩+𝐼₁+^𝑚−1_𝑘=1𝜏_𝑘≤𝑛⁺,𝑁𝑛⁺==0_{𝑛𝑢=𝑖}𝑃𝑏^⟨𝑖⟩=𝑢(1−𝜆)^𝑛−𝑢+^𝐷_𝑚=1𝛼_𝑚^𝑛−𝑚_𝑢=𝑖𝑃𝑏^⟨𝑖⟩×=𝑢_{𝑛−𝑢−𝑚+1𝑡=1}𝑃𝐼₁=𝑡×^{𝑛−𝑢−𝑡}_{𝑟=𝑚−1}𝑃^𝑚−1_𝑘=1𝜏_𝑘=𝑟×𝑃_𝑚0(𝑛−𝑢−𝑡−𝑟)⁺. (12) Taking the PGF on (12) gives 𝑝_𝑖0(𝑧)=𝑏^𝑖(𝑧)1−𝑓(𝑧)1−𝑧+𝑏^𝑖(𝑧)^𝐷_𝑚=1𝛼_𝑚𝑓^𝑚(𝑧)𝑝_𝑚0(𝑧),𝑖≥1. (13) Solving (11) and (13) yields 𝑝₀₀(𝑧)=1−𝑓(𝑧)𝑓(𝑧)𝑏(𝑧)1−𝑧1+1−∑^𝐷_𝑚=1𝛼_𝑚𝑓(𝑧)𝑏(𝑧)^𝑚,|𝑧|𝑝<1,_𝑖0(𝑧)=1−𝑓(𝑧)𝑏^𝑖(𝑧)(1−𝑧)1−∑^𝐷_𝑚=1𝛼_𝑚𝑓(𝑧)𝑏(𝑧)^𝑚,|𝑧|<1,𝑖≥1. (14)

For 1≤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 𝑃_0𝑗𝑛⁺=𝑃𝐼₁≤𝑛⁺<𝐼₁+𝑏₁,𝑁𝑛⁺=𝑗+𝑃^𝑗_𝑖=1𝜒_𝑖𝐼<𝐷,₁+𝑏₁+𝐼₂+^𝑗−1_𝑙=1𝜏_𝑙≤𝑛⁺<𝐼₁+𝑏₁+𝐼₂+^𝑗_𝑙=1𝜏_𝑙+^𝐷_𝑚=1𝑃^𝑚−1_𝑖=1𝜒_𝑖<𝐷≤^𝑚_𝑖=1𝜒_𝑖,𝐼₁+𝑏₁+𝐼₂+^𝑚−1_𝑙=1𝜏_𝑙≤𝑛⁺,𝑁𝑛⁺=𝑗. (15)

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 ^𝑛_𝑢=1𝑃𝐼₁=𝑢𝑄_𝑗(𝑛−𝑢)⁺. (16)

In the second term of (15), the “∑^𝑗_𝑖=1𝜒_𝑖<𝐷, 𝐼₁+𝑏₁+𝐼₂+∑^𝑗−1_𝑙=1𝜏_𝑙≤𝑛⁺<𝐼₁+𝑏₁+𝐼₂+∑^𝑗_𝑙=1𝜏_𝑙” implies that the first arrival occurs during the second system idle period 𝐼₂, the epoch 𝑛⁺ is located in the 𝑗th arrival-interval 𝜏_𝑗,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 𝛽_𝑗^𝑛_𝑢=𝑗+2𝑃𝐼₁+𝑏₁+𝐼₂+^𝑗−1_𝑙=1𝜏_𝑙=𝑢𝑃𝜏_𝑗>(𝑛−𝑢)⁺. (17)

Being similar to the analysis of (10), the third term of (15) is given by ^𝐷_𝑚=1𝛼_𝑚^𝑛_𝑢=𝑚+2𝑃𝐼₁+𝑏₁+𝐼₂+^𝑚−1_𝑙=1𝜏_𝑙=𝑢𝑃_𝑚𝑗(𝑛−𝑢)⁺. (18)

Multiplying (15) by 𝑧^𝑛 and summing over 𝑛 after substituting (16)–(18) into (15), we obtain, for 𝑗=1,2,…,𝐷−1, 𝑝_0𝑗(𝑧)=𝑓(𝑧)𝑞_𝑗(𝑧)+1−𝑓(𝑧)𝑓1−𝑧^𝑗+1(𝑧)𝑏(𝑧)𝛽_𝑗+𝑏(𝑧)^𝐷_𝑚=1𝛼_𝑚𝑓^𝑚+1(𝑧)𝑝_𝑚𝑗(𝑧). (19)

For 𝑖≥1, similarly, we have 𝑃_𝑖𝑗𝑛⁺=𝑃𝑏^⟨𝑖⟩>𝑛⁺,𝑁𝑛⁺=𝑗+𝛽_𝑗^𝑛_{𝑢=𝑗+𝑖}𝑃𝑏^⟨𝑖⟩+𝐼₁+^𝑗−1_𝑙=1𝜏_𝑙𝜏=𝑢×𝑃_𝑗>(𝑛−𝑢)⁺+^𝐷_𝑚=1𝛼_𝑚^𝑛_{𝑢=𝑚+𝑖}𝑃𝑏^⟨𝑖⟩+𝐼₁+^𝑚−1_𝑙=1𝜏_𝑙=𝑢×𝑃_𝑚𝑗(𝑛−𝑢)⁺. (20)

Since 𝑏^⟨𝑖⟩ can be expressed by 𝑏^⟨𝑖⟩=𝜍₁+⋯+𝜍_𝑖, where 𝜍₁,…,𝜍_𝑖 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 ^𝑖𝑛_{𝑘=1𝑢=𝑘−1}𝑃𝑏^{⟨𝑘−1⟩}=𝑢𝑄_{𝑗−𝑖+𝑘}(𝑛−𝑢)⁺. (21)

Substituting (21) into (20), multiplying (20) by 𝑧^𝑛, and summing over 𝑛 yield 𝑝_𝑖𝑗(𝑧)=^𝑖_𝑘=1𝑞_{𝑗−𝑖+𝑘}(𝑧)𝑏^𝑘−1(𝑧)+1−𝑓(𝑧)𝑓1−𝑧^𝑗(𝑧)𝑏^𝑖(𝑧)𝛽_𝑗+𝑏^𝑖(𝑧)^𝐷_𝑚=1𝛼_𝑚𝑓^𝑚(𝑧)𝑝_𝑚𝑗(𝑧). (22)

Solving (19) and (22) leads to the expressions of 𝑝_0𝑗(𝑧) and 𝑝_𝑖𝑗(𝑧); that is, for |𝑧|<1, 𝑖≥1, and 𝑗=1,2,…,𝐷−1, 𝑝_0𝑗(𝑧)=𝑓(𝑧)𝑞_𝑗(𝑧)+1−𝑓(𝑧)/(1−𝑧)𝑓^𝑗+1(𝑧)𝑏(𝑧)𝛽_𝑗+𝑓(𝑧)𝜑(𝑧)1−∑^𝐷_𝑚=1𝛼_𝑚𝑓(𝑧)𝑏(𝑧)^𝑚,𝑝_𝑖𝑗(𝑧)=^𝑖_𝑘=1𝑞_{𝑗−𝑖+𝑘}(𝑧)𝑏^𝑘−1(𝑧)+1−𝑓(𝑧)/(1−𝑧)𝑓^𝑗(𝑧)𝑏^𝑖(𝑧)𝛽_𝑗+𝑏^𝑖−1(𝑧)𝜑(𝑧)1−∑^𝐷_𝑚=1𝛼_𝑚𝑓(𝑧)𝑏(𝑧)^𝑚, (23) where 𝜑(𝑧)=∑^𝐷_𝑚=1𝛼_𝑚𝑓^𝑚(𝑧)∑^𝑚_𝑘=1𝑏^𝑘(𝑧)𝑞_{𝑗−𝑚+𝑘}(𝑧).

Finally, since 𝑁(𝑛⁺)=𝑗, 𝑗≥𝐷 indicates that the epoch 𝑛⁺ is in server busy period, by similar operations in the analysis of 𝑝_𝑖𝑗(𝑧), 𝑖≥0, 𝑗=1,2,…,𝐷−1, we obtain, for |𝑧|<1, 𝑖≥1, and 𝑗=𝐷,𝐷+1,𝐷+2,…, 𝑝_0𝑗(𝑧)=𝑓(𝑧)𝑞_𝑗(𝑧)+𝑓(𝑧)∑^𝐷_𝑚=1𝛼_𝑚𝑓^𝑚(𝑧)∑^𝑚_𝑘=1𝑏^𝑘(𝑧)𝑞_{𝑗−𝑚+𝑘}(𝑧)1−∑^𝐷_𝑚=1𝛼_𝑚𝑓(𝑧)𝑏(𝑧)^𝑚,𝑝_𝑖𝑗(𝑧)=^𝑖_𝑘=1𝑞_{𝑗−𝑖+𝑘}(𝑧)𝑏^𝑘−1(𝑧)+𝑏^𝑖−1(𝑧)∑^𝐷_𝑚=1𝛼_𝑚𝑓^𝑚(𝑧)∑^𝑚_𝑘=1𝑏^𝑘(𝑧)𝑞_{𝑗−𝑚+𝑘}(𝑧)1−∑^𝐷_𝑚=1𝛼_𝑚𝑓(𝑧)𝑏(𝑧)^𝑚. (24)

3.3. Steady-State System Size Distribution at Any Epoch 𝑛⁺

Now, based on the PGF of the transient-state system size distribution obtained above, we investigate the steady-state system size distribution at any epoch 𝑛⁺.

Let 𝑝⁺_𝑗=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 𝑝⁺_𝑗=lim^∞_{𝑛→∞𝑖=0}𝑃_𝑖𝑗𝑛⁺𝑃𝑁0⁺==𝑖_∞𝑖=0𝑃𝑁0⁺=𝑖lim_𝑛→∞𝑃_𝑖𝑗𝑛⁺=lim_𝑧↑1(1−𝑧)𝑝_𝑖𝑗(𝑧). (25) In (25), applying the expressions of 𝑝_𝑖𝑗(𝑧),𝑖,𝑗≥0, L’Hospital’s rule, and (2) leads to(1)for 𝜌=𝜆𝜇/𝜃≥1, 𝑝⁺_𝑗=0, 𝑗=0,1,2,…;(2)for 𝜌=𝜆𝜇/𝜃<1, the steady-state system size distribution {𝑝⁺_𝑗,𝑗≥0} is given by 𝑝⁺₀=1−𝜌1+∑^𝐷−1_𝑗=1𝛽_𝑗,𝑝⁺_𝑗=𝑝⁺₀𝛽_𝑗+^𝐷_𝑚=1𝛼_𝑚^𝑚_𝑘=1𝜆𝑞_{𝑗−𝑚+𝑘}(1),𝑝𝑗=1,2,…,𝐷−1,_+𝑗=𝑝⁺₀^𝐷_𝑚=1𝛼_𝑚^𝑚_𝑘=1𝜆𝑞_{𝑗−𝑚+𝑘}(1),𝑗=𝐷,𝐷+1,…, (26)where 𝑞₁(1)=1−𝐺(1−𝜆)𝜆𝐺(),𝑞1−𝜆_𝑗(1)=1𝐺()×1−𝜆_{∞∞𝑛=𝑗−1𝑘=𝑛+1}𝑔_𝑘𝐶_𝑛^𝑗−1𝜆^𝑗−1(1−𝜆)^{𝑛−𝑗+1}+^𝑗−1_𝑖=1𝑞_𝑗−𝑖(1𝐺()−)1−1−𝜆_{𝑖∞𝑚=1𝑘=𝑚}𝑔_𝑘𝐶^𝑚_𝑘𝜆^𝑚(1−𝜆)^𝑘−𝑚,𝑗=2,3,…. (27)

Remark 1. When 𝜌=𝜆𝜇/𝜃<1, noting that ∑^∞_𝑗=1∑^𝑚_𝑘=1𝑞_{𝑗−𝑚+𝑘}(1)=𝑚𝜌/(𝜆(1−𝜌)) and ∑^𝐷_𝑚=1𝑚𝛼_𝑚=1+∑^𝐷−1_𝑗=1𝛽_𝑗, we easily get ∑^∞_𝑗=0𝑝⁺_𝑗=1.
Let 𝜋⁺(𝑧)=∑^∞_𝑗=0𝑝⁺_𝑗𝑧^𝑗, |𝑧|<1 be the PGF of steady-state system size at epoch 𝑛⁺. For 𝜌=𝜆𝜇/𝜃<1, calculating directly and noting ^∞_𝑗=1𝑧^𝑗𝑚_𝑘=1𝑞_{𝑗−𝑚+𝑘}(1)=𝑧1−𝑧^𝑚1−𝐺(1−𝜆+𝜆𝑧)𝜆(𝐺(),()1−𝑧)1−𝜆+𝜆𝑧−𝑧1−𝑧1+^𝐷−1_𝑗=1𝛽_𝑗𝑧^𝑗=1−^𝐷_𝑚=1𝛼_𝑚𝑧^𝑚, (28) we have 𝜋⁺(𝑧)=1−𝜌(1−𝑧)𝐺(1−𝜆+𝜆𝑧)𝐺()⋅1−𝜆+𝜆𝑧−𝑧1+∑^𝐷−1_𝑗=1𝛽_𝑗𝑧^𝑗1+∑^𝐷−1_𝑗=1𝛽_𝑗. (29)

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: 𝑃Υ₁=1=01+∑^𝐷−1_𝑗=1𝛽_𝑗,𝑃Υ₁=𝛽=𝑗^𝑗1+∑^𝐷−1_𝑗=1𝛽_𝑗,𝑗=1,2,…,𝐷−1. (30)

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 by 𝐸[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 𝐸𝐿⁺𝜆=𝜌+²𝐺^′′(1)2+∑1−𝜌^{𝐷−1𝑗=1}𝑗𝛽_𝑗1+∑^𝐷−1_𝑗=1𝛽_𝑗, (31) where 𝑗≤0, ∑^𝑗_𝑖=1=0.

3.4. Steady-State System Size Distributions at Epochs 𝑛 and 𝑛⁻

Let 𝑃_𝑖𝑗(𝑛)=𝑃{𝑁(𝑛)=𝑗∣𝑁(0)=𝑖} and 𝑃_𝑖𝑗(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⁻, 𝑝_𝑗=lim_𝑛→∞𝑃{𝑁(𝑛)=𝑗} and 𝑝⁻_𝑗=lim_𝑛→∞𝑃{𝑁(𝑛⁻)=𝑗}, 𝑗=0,1,2,… denote steady- state system size distributions at epochs 𝑛 and 𝑛⁻, 𝜋(𝑧)=∑^∞_𝑗=0𝑝_𝑗𝑧^𝑗, 𝜋⁻(𝑧)=∑^∞_𝑗=0𝑝⁻_𝑗𝑧^𝑗, |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 𝑃_𝑖𝑗(𝑛⁻)=𝑃𝑁(𝑛⁻)=𝑗∣𝑁(0⁻)𝑁()=𝑖=𝑃𝑛−1⁺=𝑗∣𝑁0⁺=𝑖=𝑃_𝑖𝑗(𝑛−1)⁺. (32) Taking limit as 𝑛→∞ under the stability condition, 𝜌=𝜆𝜇/𝜃<1, we get 𝑝⁻_𝑗=𝑝⁺_𝑗𝜋,𝑗≥0,whichisequivalentto₋(𝑧)=𝜋⁺(𝑧),|𝑧|<1. (33)

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)𝑝⁻₀, 𝑝_𝑗=(1−𝜆)𝑝⁻_𝑗+𝜆𝑝⁻_𝑗−1, 𝑗=1,2,…. By using (33), 𝜋⁺(𝑧) in (29), and 𝐸(𝐿⁺) in (31), when 𝜌=𝜆𝜇/𝜃<1, we have 𝑝₀=(1−𝜆)𝑝⁺₀,𝑝_𝑗=(1−𝜆)𝑝⁺_𝑗+𝜆𝑝⁺_𝑗−1,𝑗=1,2,…,𝜋(𝑧)=(1−𝜆+𝜆𝑧)1−𝜌(1−𝑧)𝐺(1−𝜆+𝜆𝑧)𝐺()⋅1−𝜆+𝜆𝑧−𝑧1+∑^𝐷−1_𝑗=1𝛽_𝑗𝑧^𝑗1+∑^𝐷−1_𝑗=1𝛽_𝑗,𝐸(𝐿)𝜆=𝜆+𝜌+²𝐺^′′(1)2+∑1−𝜌^{𝐷−1𝑗=1}𝑗𝛽_𝑗1+∑^𝐷−1_𝑗=1𝛽_𝑗. (34)(35)(36)

Remark 4. At epochs 𝑛⁻, 𝑛⁺, and 𝑛, the following relations are clear:(1)𝑝⁻_𝑗=𝑝⁺_𝑗≠𝑝_𝑗, 𝑗=0,1,2,…, or 𝜋⁻(𝑧)=𝜋⁺(𝑧)≠𝜋(𝑧);(2)𝑝⁻₀=𝑝⁺₀>𝑝₀, 𝐸(𝐿⁻)=𝐸(𝐿⁺)<𝐸(𝐿).

4. Special Cases

Case 1. Let 𝜃=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 𝑝⁺₀=1−𝜌,𝑝⁺_𝑗=𝜆1−𝜌𝑞_𝑗(1),𝑗=1,2,…,𝜋⁺(𝑧)=1−𝜌(1−𝑧)𝐺(1−𝜆+𝜆𝑧)𝐺(),1−𝜆+𝜆𝑧−𝑧𝐸𝐿⁺𝜆=𝜌+²2𝐺1−𝜌^′′(1), (37)(38)(39) where 𝑞_𝑗(1),𝑗=1,2,…, are determined by (7), in which 𝑔_𝑗=𝑔_𝑗 and 𝐺(𝑧)=𝐺(𝑧).

Case 2. Let 𝜃=1, 𝐷=1, and the service times 𝜒_𝑖,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 𝑝_𝑗^(𝑒)=1−𝜌1−𝜆𝜌−𝜆1−𝜆^𝑗𝜆,𝑗=0,1,…,if𝜌=𝛿<1, (40) which agrees with that in [24].

In fact, denote by 𝑁^(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 {𝑝_𝑗^(𝑒)} and {𝑝_𝑗^(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 𝑝⁺_𝑗=𝑝_𝑗^(𝑒)+,𝑗≥0. Since the customer arrival is a Bernoulli process with rate 𝜆, we have 𝑝⁺₀=𝑝₀^(𝑒)+=(1−𝜆)𝑝₀^(𝑒),𝑝⁺_𝑗=𝑝_𝑗^(𝑒)+=(1−𝜆)𝑝_𝑗^(𝑒)+𝜆𝑝^(𝑒)_𝑗−1,𝑗≥1. (41) It follows from (41) that, for 𝜌=𝜆/𝛿<1 and |𝑧|<1, the relation between 𝜋⁺(𝑧)=∑^∞_𝑗=0𝑝⁺_𝑗𝑧^𝑗 and 𝜋^(𝑒)(𝑧)=∑^∞_𝑗=0𝑝_𝑗^(𝑒)𝑧^𝑗 is 𝜋⁺(𝑧)=(1−𝜆+𝜆𝑧)𝜋^(𝑒)(𝑧). (42) Using 𝜋⁺(𝑧) in (38) and noting that the service times obey geometric distribution with mean 1/𝛿, we obtain 𝜋⁺(𝑧)=1−𝜌(1−𝜆+𝜆𝑧).1−𝜆+𝜆𝑧−𝜌𝑧 (43) From (42) and (43), we get ^∞_𝑗=0𝑝_𝑗^(𝑒)𝑧^𝑗=𝜋^(𝑒)(𝑧)=1−𝜌=1−𝜆+𝜆𝑧−𝜌𝑧1−𝜌1−𝜆^∞_𝑗=0𝜌−𝜆1−𝜆^𝑗𝑧^𝑗. (44) 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 Section 1. 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 Section 3, based on the results obtained in (26) and (34), we numerically get the steady-state radio signal number distributions {𝑝_𝑗,𝑗≥0} and {𝑝⁺_𝑗,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 epochs 𝑛 and 𝑛⁺ for a wireless local area network (𝜆=0.25, 𝜇=1, 𝜃=0.4, and 𝐷=20).

Figure 1 

Steady-state radio signal number distributions at epochs 𝑛 and 𝑛+ for a wireless local area network.

Remark 5. With the software of Matlab 7.0, the data in Table 1 are accurate to five decimal places.
From the two curves of Figure 1, we conclude that the steady-state distribution probabilities {𝑝_𝑗,𝑗≥0} and {𝑝⁺_𝑗,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 𝑝_𝑗 and 𝑝⁺_𝑗 have a small difference. Since 𝑝⁺_𝑗 or 𝑝_𝑗,𝑗>20 tends to zero, it is improper for this WLAN to design an enough large space capacity.
Let 𝐾 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-policy Geo^𝑋/G/1 queue with feedback and vacation, will be investigated by the similar analysis techniques.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referees and editor for their valuable comments and suggestions. This work is supported by the Basic and Frontier Research Foundation of Chongqing of China (cstc2013jcyjA00008) and the Scientific Research Starting Foundation for Doctors of Chongqing University of Technology (2012ZD48).