Analysis of an M/M/c Queueing System with Impatient Customers and Synchronous Vacations

Abstract

We consider an M/M/c queueing system with impatient customers and a synchronous vacation policy, where customer impatience is due to the servers’ vacation. Whenever a system becomes empty, all the servers take a vacation. If the system is still empty, when the vacation ends, all the servers take another vacation; otherwise, they return to serve the queue. We develop the balance equations for the steady-state probabilities and solve the equations by using the probability generating function method. We obtain explicit expressions of some important performance measures by means of the two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. We derive closed-form expressions of some important performance measures for two special cases. Finally, some numerical results are also presented.

1. Introduction

Queueing systems with vacations have been developed for wide range of applications in flexible manufacturing and computer communication systems over more than two decades. Literatures relating to this topic can be found in several excellent surveys by Doshi [1], Takagi [2], and a monograph by Tian and Zhang [3]. There is now growing interest in the analysis of queueing systems with impatient customers. This is due to their potential applications in communication systems, call centers, production-inventory systems, and many other related areas; see for instance [4–6] and the references therein.

The M/M/𝑐 queue with exponentially distributed vacation times was first studied by Levy and Yechiali [7], where each server takes vacations independently when the server finishes a service and finds no customers waiting in the queue. In the literature, this type of vacation policy is called asynchronous vacation policy. Vinod [8] studied this model by using QBD process and obtained the matrix-geometric solution. Chao and Zhao [9] investigated the multiserver vacation models of both synchronous (servers taking the same vacation together) and asynchronous types and provided some algorithms for computing the stationary probability distributions and expected performance measures. Tian et al. [10] and Zhang and Tian [11] have established the conditional stochastic decomposition properties on queue length and waiting time and provided the stationary distributions for the queue length and the waiting time in the Markovian multiserver synchronous vacation queueing systems.

Recently, Altman and Yechiali [12] presented a comprehensive analysis for M/M/𝑐 queueing models with server vacations and customer impatience, where customers became impatient only when the servers were on vacation. They considered asynchronous vacation policy for both the single and the multiple vacation cases by using the probability generating function method. However, they did not obtain the detailed results for the stationary probabilities and some expected performance measures such as the mean queue length and expected waiting time. For other works on the vacation queueing models with impatient customers, we refer to Altman and Yechiali [13], Yue et al. [14], Perel and Yechiali [15], and Economou and Kapodistria [16].

To the best of our knowledge, there is no work on multiserver synchronous vacation queueing models with impatient customers. In this paper, we consider an M/M/𝑐 queueing system with impatient customers and synchronous vacations. When a server finishes serving a customer and finds the system empty, all servers immediately leave for a vacation. If all servers return from a vacation to find an empty queue, they immediately leave for another vacation; otherwise, all servers return to serve the queue. This type of vacation is called a multiple synchronous vacation policy. Customers became impatient only when the servers were on vacation. That is, an arriving customer who finds that all servers are on vacation activates an “impatience timer.” If the customer’s service has not been completed before the customer’s impatience timer expires, the customer abandons the queue and never returns.

The model considered in this paper has potential applications in practical systems. For example, consider a production-inventory system with impatient customers, where a single product is produced at a multipurpose facility. The major job of the facility is to produce products to fulfill customer orders. In addition, the facility may perform other optional jobs utilizing the time between subsequent productions. The production facility can produce ahead of the demand in a make-to-stock fashion. However, the system manager does not want to keep higher level of inventory of items because more items in inventory result in the increasing of the holding costs. Therefore, whenever the last order is completed and no order occurs, the manager may decide to stop the major production and perform optional jobs for a period of time. Upon completion of each optional job the manager checks the orders and decides whether or not to restart major production. If no orders occur at this moment, a decision is made to perform other optional jobs next. The optional jobs can be referred to as a sequence of finite maintenance policy (such as inspection, replacement, or preventive maintenance for machines in the facility) or secondary jobs. Upon arrival, an order is either fulfilled from the inventory if any is available or back-ordered. Customers whose orders are back-ordered may become impatient and decide to cancel their orders if the customers’ waiting time exceeds a customer’s level of patience. This is especially likely when the facility performs optional jobs. This system can be modeled by our model developed in this paper. For other similar examples in communication networks, we refer to Ke [17] and Ke and Chang [18].

The rest of paper is organized as follows. In Section 2, we describe the model. In Section 3, we first develop the differential equations for the probability generating functions of the steady-state probabilities. Then, we obtain the explicit expressions for some performance measures in terms of two indexes. Based on these, we obtain some results about limiting behavior for some performance measures. The closed-form expressions of some performance measures are obtained for two special cases. In Section 4, we present some numerical results. Conclusions are given in Section 5.

2. Model Description

We consider an M/M/𝑐 queueing system with impatient customers and a synchronous vacation policy. Customers arrive according to a Poisson process at rate 𝜆. The service is provided by 𝑐 servers, who serve the customers on a first-come first-served (FCFS) basis. The service time of each customer is exponentially distributed with mean 1/𝜇.

The multiple synchronous vacation policy is described as follows. When the server finishes serving a customer and finds the system empty, all servers immediately leave for a vacation. If servers return from a vacation to find an empty queue, they immediately leave for another vacation; otherwise, they return to serve the queue. The duration of a vacation is exponentially distributed with mean 1/𝛾.

During the vacation, customers are impatient. That is, an arriving customer who finds that all servers are on vacation activates an “impatience timer" 𝑇, which is exponentially distributed with mean 1/𝜉. If the customer’s service has not been completed before the customer’s timer expires, the customer abandons the queue and never returns.

3. Stationary Analysis

In this section, we present a stationary analysis for the model described in the last section.

3.1. Balance Equations

Let 𝐿(𝑡) denote the number of customers in the system at time 𝑡 and let 𝐽(𝑡) denote the status of the server at time 𝑡, which is defined as follows: 𝐽(𝑡)=0 denotes that all 𝑐 servers are taking a vacation at time 𝑡 and 𝐽(𝑡)=1 denotes that some servers are busy serving customers at time 𝑡. Then, the process {(𝐿(𝑡),𝐽(𝑡)),𝑡≥0} defines a continuous-time Markov process with state space Ω={(0,0)}∪{(𝑛,𝑗)∶𝑛≥1,𝑗=0,1}.

Let 𝑃_𝑛𝑗=lim_𝑡→∞𝑃𝐿(𝑡)=𝑛,𝐽(𝑡)=𝑗,𝑛,𝑗∈Ω, (1) denote the steady-state probabilities of the process {(𝐿(𝑡),𝐽(𝑡)),𝑡≥0}. Then, the set of balance equations is given as follows: 𝜆𝑃₀₀=𝜇𝑃₁₁+𝜉𝑃₁₀,𝜆+𝑛𝜉+𝛾𝑃_𝑛0=𝜆𝑃_(𝑛−1)0+(𝑛+1)𝜉𝑃_(𝑛+1)0,𝑛≥1,𝜆+𝜇𝑃₁₁=2𝜇𝑃₂₁+𝛾𝑃₁₀,𝜆+𝑛𝜇𝑃_𝑛1=𝜆𝑃_(𝑛−1)1+(𝑛+1)𝜇𝑃_(𝑛+1)1+𝛾𝑃_𝑛0,2≤𝑛≤𝑐−1,𝜆+𝑐𝜇𝑃_𝑛1=𝜆𝑃_(𝑛−1)1+𝑐𝜇𝑃_(𝑛+1)1+𝛾𝑃_𝑛0,𝑛≥𝑐, (2)(3)(4)(5)(6) and the normalizing condition is as follows: ^∞_𝑛=0𝑃_𝑛0+^∞_𝑛=1𝑃_𝑛1=1. (7)

3.2. Generating Functions

We define the probability generating functions as follows: 𝐺₀(𝑧)=^∞_𝑛=0𝑃_𝑛0𝑧^𝑛,𝐺₁(𝑧)=^∞_𝑛=1𝑃_𝑛1𝑧^𝑛,0≤𝑧≤1. (8)

Then, multiplying (3) by 𝑧^𝑛, summing all possible values of 𝑛, and using (2), we get 𝜉()𝑑1−𝑧𝐺𝑑𝑧⁰(𝑧)=𝜆(1−𝑧)+𝛾𝐺₀(𝑧)−𝛾𝑃₀₀+𝜇𝑃₁₁. (9) Similarly, multiplying (5) and (6) by 𝑧^𝑛, then summing all possible values of 𝑛, and using (4), we get (1−𝑧)𝜆𝑧−𝜇𝐺₁(𝑧)=𝛾𝑧𝐺₀(𝑧)−𝛾𝑃₀₀+𝜇𝑃₁₁𝑧+𝜇(1−𝑧)^𝑐_𝑛=1(𝑛−𝑐)𝑃_𝑛1𝑧^𝑛. (10)

The differential equation (9) is the same as (2.4) in the paper written by Altman and Yechiali [12], where they solve this differential equation and obtain 𝐺₀(𝑧) as follows: 𝐺₀(𝑧)=𝑒^{(𝜆/𝜉)𝑧}(1−𝑧)^−𝛾/𝜉×𝐺₀(0)−𝐻𝜉^𝑧₀(1−𝑥)^{(𝛾/𝜉)−1}𝑒^{−(𝜆/𝜉)𝑥}𝑑𝑥, (11) where 𝐻=𝛾𝑃₀₀+𝜇𝑃₁₁. (12) See Altman and Yechiali [12, Equation (2.11)]. Then 𝐺₀(1)=𝑒^{(𝜆/𝜉)𝑧}𝐺₀(0)−𝐻𝜉¹₀(1−𝑥)^{(𝛾/𝜉)−1}𝑒^{−(𝜆/𝜉)𝑥}𝑑𝑥×lim_𝑧→1(1−𝑧)^−𝛾/𝜉. (13) Since 𝐺₀(1)=∑^∞_𝑛=0𝑃_𝑛0>0 and lim_𝑧→1(1−𝑧)^−𝛾/𝜉=∞, we must have that 𝑃₀₀=𝐺₀(0)=𝐻𝜉𝐾, (14) where 𝐾=¹₀𝑒^{−(𝜆/𝜉)𝑥}(1−𝑥)^{(𝛾/𝜉)−1}𝑑𝑥. (15) Substituting (14) into (11) and noting that 𝐺₀(0)=𝑃₀₀, we have 𝐺₀(𝑧)=𝑒^{(𝜆/𝜉)𝑧}(1−𝑧)^𝛾/𝜉11−𝐾^𝑧₀(1−𝑥)^{(𝛾/𝜉)−1}𝑒^{−(𝜆/𝜉)𝑥}𝑑𝑥𝑃₀₀. (16)

Equation (10) can be written as follows: 𝐺₁(𝑧)=𝛾𝐺₀(𝑧)−𝐻𝑧()−𝜇𝜆𝑧−𝜇1−𝑧𝑄(𝑧),𝜆𝑧−𝜇 (17) where 𝑄(𝑧)=∑^𝑐_𝑛=1(𝑐−𝑛)𝑃_𝑛1𝑧^𝑛.

Equation (16) shows that 𝐺₀(𝑧) can be expressed in terms of 𝑃₀₀. Equation (17) shows that 𝐺₁(𝑧) can be expressed in terms of 𝐺₀(𝑧), 𝐻, and 𝑄(𝑧). In other words, once 𝑃₀₀ and 𝑃_𝑗1 (𝑗=1,2,…,𝑐−1) are obtained, 𝐺₀(𝑧) and 𝐺₁(𝑧) are completely determined.

3.3. Performance Measures

In this subsection, we derive some performance measures of the system by using the expressions of the PGF we obtained in last section. Furthermore, we consider the limiting behavior for some performance measures.

3.3.1. Expected System Sizes

Let 𝐿₀ and 𝐿₁ be the system size when all servers are on vacation and not on vacation, respectively. Then, the expected system sizes 𝐸[𝐿₀] and 𝐸[𝐿₁] are defined by 𝐸𝐿_𝑗=^∞_𝑛=1𝑛𝑃_𝑛𝑗,𝑗=0,1. (18) Let 𝐺^′₀(1)=𝑑𝐺𝑑𝑧⁰(𝑧)_𝑧=1,𝐺₀^′′(1)=𝑑²𝑑𝑧²𝐺₀(𝑧)_𝑧=1. (19) Then, 𝐺^′₀(1)=𝐸[𝐿₀] and 𝐺₀^′′(1)=𝐸[𝐿₀(𝐿₀−1)].

First, we derive 𝐸[𝐿₀]. Using L’Hôpital rule, we have from (16) that 𝐺₀(1)=𝜉𝑃𝛾𝐾⁰⁰. (20) Substituting (14) into (20), we get 𝛾𝐺₀(1)=𝐻. (21) From (17), using L’Hôpital rule, we get 𝐺₁(1)=lim_𝑧→1𝐺₁(𝑧)=𝛾𝐺₀(1)−𝐻+𝛾𝐺^′₀(1)+𝜇𝜇−𝜆𝑄(1),𝜇−𝜆 (22) where 𝑄(1)=∑^𝑐_𝑗=1(𝑐−𝑗)𝑃_𝑗1, and substituting (21) into (22), we get 𝐺₁(1)=𝛾𝐸𝐿𝜇−𝜆⁰+𝜇𝑄(1).𝜇−𝜆 (23) From (9), we get 𝐸𝐿₀=lim_𝑧→1𝐺^′₀(𝑧)=lim_𝑧→1𝜆(1−𝑧)+𝛾𝐺₀(𝑧)−𝐻𝜉()=1−𝑧−𝜆𝐺₀(1)+𝛾𝐺^′₀(1)=−𝜉−𝜆𝐺₀(1)+𝛾𝐸𝐿₀,−𝜉 (24) where the third equality follows by using L’Hôpital rule. Thus, we have 𝐺₀(1)=𝛾+𝜉𝜆𝐸𝐿₀. (25) Using (22) and (25) and noting that 𝐺₀(1)+𝐺₁(1)=1, we obtain 𝐸𝐿₀=𝜆1−𝜌−𝜆𝛾+𝜉1−𝜌𝑄(1),𝛾+𝜉1−𝜌 (26) where 𝜌=𝜆/𝜇.

Now, we derive 𝐸[𝐿₁]. From (17), we have 𝐸𝐿₁=lim_𝑧→1𝐺^′₁(𝑧)=𝛾2(𝜇−𝜆)²lim_𝑧→1𝜇−𝜆𝑧𝑧𝐺₀^′′(𝑧)+2𝐺^′₀(𝑧−𝜆)(𝜇−𝜆)²lim_𝑧→1𝑧𝛾𝐺₀(𝑧)−𝐻(1−𝑧)+𝜇lim_𝑧→1𝜇−𝜆𝑧𝑄^′(𝑧)+𝜆𝑄(𝑧)(𝜇−𝜆𝑧)²=𝛾𝜇−𝜆𝐺₀^′′(1)+2𝜇𝛾𝐺^′₀(1)2(𝜇−𝜆)²+𝜇𝜇−𝜆𝑄^′(1)+𝜆𝑄(1)(𝜇−𝜆)²=𝜇−𝜆𝛾𝐸𝐿₀𝐿₀−1+2𝜇𝛾𝐸𝐿₀2(𝜇−𝜆)²+1𝑄1−𝜌^′(1)+𝜌(1−𝜌)²𝑄(1), (27)

where 𝑄^′(1)=𝑑𝑄(𝑧)𝑑𝑧_𝑧=1=^𝑐−1_𝑗=1𝑗𝑐−𝑗𝑃_𝑗1. (28)

In order to obtain 𝐺₀^′′(1), we take derivatives for two times on both sides of (9); then we get 𝜉(1−𝑧)𝐺₀^′′′(𝑧)+2𝜆𝐺^′₀(𝑧)=𝜆(1−𝑧)+𝛾+2𝜉𝐺₀^′′(𝑧), (29) where 𝐺₀^′′′(𝑧)=𝑑³/𝑑𝑧³𝐺₀(𝑧). Letting 𝑧=1 in (29), we get 𝐺₀^′′(1)=2𝜆𝐺𝛾+2𝜉^′0(1) (30) or, equivalently, 𝐸𝐿₀𝐿₀−1=2𝜆𝐸𝐿𝛾+2𝜉⁰. (31) Substituting (31) into (27), we get 𝐸𝐿₁=𝜌𝛾1−𝜌+1𝛾+2𝜉𝜆𝐸𝐿1−𝜌⁰+1𝑄1−𝜌^′(1)+𝜌(1−𝜌)²𝑄(1). (32)

Let 𝐿 be the number of customers in the system. Then, the expected system size can be calculated by 𝐸[𝐿]=𝐸[𝐿₀]+𝐸[𝐿₁], where 𝐸[𝐿₀] and 𝐸[𝐿₁] are given by (26) and (32).

Remark 1. If we take derivatives for (𝑘+1)th times on both sides of (9) and use the same method as getting (31), then we get 𝑑^𝑘𝐺₀(𝑧)𝑑𝑧^𝑘_𝑧=1=𝑘𝜆×𝑑𝛾+𝑘𝜉^𝑘−1𝐺₀(𝑧)𝑑𝑧^𝑘−1_𝑧=1 (33) or, equivalently, 𝐸𝐿₀𝐿₀−1⋯𝐿₀=−𝑘𝑘𝜆𝐸𝐿𝛾+𝑘𝜉⁰𝐿₀−1⋯𝐿₀−𝑘+1. (34) Using this iteration, we get 𝐸𝐿₀𝐿₀−1⋯𝐿₀−𝑘=^𝑘+1_𝑖=2𝑖𝜆𝐸𝐿𝛾+𝑖𝜉⁰, (35) for 𝑘=1,2,….

3.3.2. Other Performance Measures

Now, we derive some other performance measures such as the probability when servers are on vacation (or not on vacation), the proportion of customers served per unit of time, and the average rate of abandonment due to impatience.

Let 𝑃_.0=∑^∞_𝑛=0𝑃_𝑛0 and 𝑃_.1=∑^∞_𝑛=1𝑃_𝑛1. Clearly, 𝑃_.0=𝐺₀(1) is the probability when servers are on vacation and 𝑃_.1=𝐺₁(1)=1−𝑃_.0 is the probability when servers are not on vacation.

Substituting (26) into (23) and (25), we get 𝑃_.0=1−𝜌𝛾+𝜉−𝛾+𝜉1−𝜌𝛾+𝜉𝑄(1),𝛾+𝜉1−𝜌𝑃_.1=𝜌𝛾+𝛾+𝜉1−𝜌𝛾+𝜉𝑄(1).𝛾+𝜉1−𝜌 (36)(37)

When the system is in state (𝑛,1), the service rates of the servers are 𝑛𝜇 for 𝑛=1,2,…,𝑐 and 𝑐𝜇 for 𝑛=𝑐+1,𝑐+2,…, respectively. Thus, the expected number of customers served per unit of time is given by 𝑁_𝑠=^𝑐_𝑛=1𝜇𝑛𝑃_𝑛1+^∞_𝑛=𝑐+1𝑐𝜇𝑃_𝑛1=𝜇𝑐𝑃_.1−𝑄(1), (38) implying that the proportion of customers served per unit of time is given by 𝑃_𝑠=𝑁_served𝜆=1𝜌𝑐𝑃_.1−𝑄(1), (39) where 𝑃_.1 is given by (37).

When the system is in state (𝑛,0), 𝑛≥1, the rate of abandonment of a customer due to impatience is 𝑛𝜉. Thus, the average rate of abandonment due to impatience is given by 𝑅_𝑎=^∞_𝑛=1𝑛𝜉𝑃_𝑛0=𝜉𝐸𝐿₀, (40) where 𝐸[𝐿₀] is given by (26).

All the performance measures of the system we derived in this subsection are expressed in terms of 𝑄(1) or/and 𝑄^′(1). In the next subsection, we consider the calculation of these two indexes 𝑄(1) and 𝑄^′(1).

Remark 2. Let 𝑃_𝐼=∑^𝑐−1_𝑗=1𝑃_𝑗1 be the probability that there are idle servers when all the servers are not on vacation. Clearly, 𝑄(1)/𝑃_𝐼 is the conditional expected number of idle servers provided that there are idle servers when all servers are not on vacation and 𝑄^′(1)/𝑃_𝐼 is the conditional expectation of the product of the number of the busy servers and the number of the idle servers provided that there are idle servers when all servers are not on vacation.

3.3.3. Limiting Behavior

We consider the limiting behavior for some performance measures when 𝜌→1.

Since 𝑃_.0≥0, from (36), it is easy to see that 0≤𝑄(1)≤1−𝜌, which implies that lim_𝜌→1𝑄(1)=0. (41) Since 𝑄(1)=∑^𝑐_𝑗=1(𝑐−𝑗)𝑃_𝑗1, it is easy to see that lim_𝜌→1𝑃_𝑗1=0 for 𝑗=1,2,…,𝑐−1. This implies that lim_𝜌→1𝑄^′(1)=lim^𝑐_{𝜌→1𝑗=1}𝑗𝑐−𝑗𝑃_𝑗1=0. (42)

The two curves of 𝑄(1) and 𝑄^′(1) with respect to 𝜌 are shown in Figure 1, where the parameters are given as follows: 𝜉=1, 𝛾=0.5, 𝜇=2, and 𝑐=5. From Figure 1, it is observed that both 𝑄(1) and 𝑄^′(1) go to zero when 𝜌→1. This observation coincides with the results given by (41) and (42). Furthermore, Figure 1 shows that the two curves are very close when 𝜌 is very small.

Figure 1 

Effects of 𝜌 on 𝑄(1) and 𝑄′(1).

Noting (41), we get from (36) and (37) that lim_𝜌→1𝑃_.0=0,lim_𝜌→1𝑃_.1=1. (43) Further, we get from (39) that lim_𝜌→1𝑃_𝑠=𝑐. (44)

Remark 3. Noting that lim_𝜌→1𝑃_𝑗1=0 for 𝑗=1,2,…,𝑐−1 and using (43), we get lim_𝜌→1∑^∞_𝑗=𝑐𝑃_𝑗1=1. This implies that if 𝜌→1, then all servers in the system will be busy. This explains the result given by (44).

In Figures 2 and 3, we investigate the effect of 𝜌 on 𝑃_.0 and 𝑃_𝑠, where the parameters are given as follows: 𝜉=0.3, 𝛾=0.5, and 𝜇=2.

Figure 2 

Effect of 𝜌 on 𝑃.0.

Figure 3 

Effect of 𝜌 on 𝑃𝑠.

From Figure 2, it is observed that 𝑃_.0 is a decreasing function of 𝜌 and it has its limit at 0 when 𝜌→1 for all three values of 𝑐. This agrees with (43). The three curves show that 𝑃_.0 has its limit at 1 when 𝜌→0. This agrees with the intuitive expectation. In addition, we observe that 𝑃_.0 is a concave function of 𝜌 for 𝑐=1, but it is not a concave function of 𝜌 for other values of 𝑐=5 and 𝑐=10.

From Figure 3, it is observed that 𝑃_𝑠 is an increasing function of 𝜌 for 𝑐=1 which agrees with the analytical result given by Altman and Yechiali [12, page 270, Equation (3.32)], but it is not an increasing function of 𝜌 for other values of 𝑐=5 and 𝑐=10. All the three curves show that 𝑃_𝑠 has its limit at 𝑐 when 𝜌→1. This agrees with (44).

Noting (41), we have from (26) that lim_𝜌→11𝜆𝐸𝐿₀=lim_𝜌→11𝜇𝐸𝐿₀=0. (45) Further, we have from (32) that lim_𝜌→1𝐸𝐿₁=lim_𝜌→1𝛾𝜇𝜆1−𝜌+1𝛾+2𝜉𝐸𝐿1−𝜌⁰. (46) Thus, when 𝜌→1 we have an approximation for 𝐸[𝐿₁] as follows: 𝐸𝐿₁≅𝛾𝜇𝜆1−𝜌+1𝛾+2𝜉𝐸𝐿1−𝜌⁰. (47)

Remark 4. The right hand side of (47) is the same as the corresponding expected system size for the single server model (see Altman and Yechiali [12, page 264]).

In Figures 4 and 5, we investigate the effect of 𝜌 on 𝐸[𝐿₀] and 𝐸[𝐿₁], where the parameters are given as follows: 𝜉=0.3, 𝛾=0.5, and 𝜇=2.

Figure 4 

Effect of 𝜌 on 𝐸[𝐿0].

Figure 5 

Effect of 𝜌 on 𝐸[𝐿1].

From Figure 4, it is observed that 𝐸[𝐿₀] is a concave function of 𝜌 for all three values of 𝑐. It is insensitive to the variables of 𝑐 when 𝜌 is very small or 𝜌 is close to 1. From Figure 5, it is observed that 𝐸[𝐿₁] is an increasing convex function of 𝜌. Figure 5 shows that 𝐸[𝐿₁] increases slowly when 𝜌 is less than 0.9, but it increases significantly when 𝜌 is larger than 0.9. In addition, we observe that all three curves almost coincide when 𝜌→1. This agrees with (47) (see Remark 4).

3.4. Calculation of 𝑄(1) and 𝑄^′(1)

In order to compute 𝑄(1) and 𝑄^′(1), we need to compute 𝑃_𝑗1 for 𝑗=1,2,…,𝑐−1. From (2), (3), (4), and (5), the 2𝑐−1 unknown probabilities 𝑃_𝑗1 for 𝑗=1,2,…,𝑐−1 and 𝑃_𝑗0 for 𝑗=0,1,…,𝑐−1 satisfy the following 2𝑐−3 linear equations:𝜆𝑃₀₀=𝜇𝑃₁₁+𝜉𝑃₁₀,𝜆+𝑛𝜉+𝛾𝑃_𝑛0=𝜆𝑃_(𝑛−1)0+(𝑛+1)𝜉𝑃_(𝑛+1)0,1≤𝑛≤𝑐−2,𝜆+𝜇𝑃₁₁=2𝜇𝑃₂₁+𝛾𝑃₁₀,𝜆+𝑛𝜇𝑃_𝑛1=𝜆𝑃_(𝑛−1)1+(𝑛+1)𝜇𝑃_(𝑛+1)1+𝛾𝑃_𝑛0,2≤𝑛≤𝑐−2. (48)(49)(50)(51) Therefore, we need another two independent equations to calculate all 2𝑐−1 unknowns.

We show that 𝑃₁₁ can be expressed by 𝑃₀₀. Equation (14) can be written as 𝐺₀(0)=𝑃₀₀=𝛾𝑃₀₀+𝜇𝑃₁₁𝜉𝐾 (52) which yields that 𝑃₁₁=𝛿𝑃₀₀, (53) where 𝛿=𝜉−𝛾𝐾.𝐾𝜇 (54)

Remark 5. It is easy to confirm that 𝜉−𝛾𝐾>0 (see Altman and Yechiali [12, page 263]).

Substituting (36) into (21), we get 𝑃₀₀+𝜇𝛾𝑃₁₁=𝜇−𝜆𝛾+𝜉−𝜇𝜇𝛾+𝜉𝜇−𝜆𝛾+𝜉𝑄(1).𝜇𝛾+𝜉𝜇−𝜆 (55)

Equations (53) and (55) give another two independent equations. All in all, we have 2𝑐−1 independent equations to solve for the 2𝑐−1 unknowns. In the following, we solve these equations analytically.

Substituting (53) into (48) and (50), we have 𝜆−𝜇𝛿𝑃₀₀=𝜉𝑃₁₀,𝜆+𝜇𝛿𝑃₀₀=2𝜇𝑃₂₁+𝛾𝑃₁₀. (56)(57) Thus, 𝑃_𝑗0, 𝑗=1,2,…,𝑐−1, and 𝑃_𝑗1, 𝑗=2,3,…,𝑐−1, satisfy (49), (51), (56), and (57). These equations can be written as equations in matrix form.

For this, we define the following column vectors: 𝑃₀=(𝑃₁₀,𝑃₂₀,…,𝑃_(𝑐−1)0)^𝑇,𝑃₁=(𝑃₂₁,𝑃₃₁,…,𝑃_(𝑐−1)1)^𝑇, (58) Then, we have 𝐴𝑃₀=𝐷𝑃₀₀,𝑃₀+𝐶𝑃₁=𝐸𝑃₀₀, (59)

where 𝐴, 𝐵, and 𝐶 are matrices given as follows: 𝐴=𝜉00⋯000−𝑎₁2𝜉0⋯000𝜆−𝑎₂3𝜉⋯000⋮⋮⋮⋮⋮⋮000⋯𝜆−𝑎_𝑐−2()𝜉,,𝑐−1𝐵=𝛾0⋯000𝛾⋯00⋮⋮⋮⋮00⋯𝛾0𝐶=2𝜇00⋯000−𝑏₂3𝜇0⋯000𝜆−𝑏₃4𝜇⋯000⋮⋮⋮⋮⋮⋮000⋯𝜆−𝑏_𝑐−2()𝜇,𝑐−1 (60) where 𝑎_𝑛=𝜆+𝑛𝜉+𝛾 and 𝑏_𝑛=𝜆+𝑛𝜇 for 𝑛=1,2,…,𝑐−2 and 𝐷 and 𝐸 are two column vectors given as follows: 𝐷=(𝜆−𝜇𝛿,−𝜆,0,…,0)^𝑇,𝐸=((𝜆+𝜇)𝛿,−𝜆𝛿.0,…,0)^𝑇. (61)

Clearly, matrices 𝐴 and 𝐶 are inverse matrices. Thus, from (59), we have 𝑃₀=𝐴⁻¹𝐷𝑃₀₀,𝑃₁=𝐶⁻¹𝐸−𝐵𝐴⁻¹𝐷𝑃₀₀, (62) where 𝐴⁻¹ and 𝐶⁻¹ are inverses of matrices 𝐴 and 𝐶, respectively.

Let 𝑒₀ be a vector with 𝑐−1 elements all to be one and let 𝑒₁ be a column vector with 𝑐−2 elements all to be one. Using (53) and (62), 𝑄(1) can be written by 𝑄(1)=(𝑐−1)𝛿𝑃₀₀+𝐹𝐶⁻¹𝐸−𝐵𝐴⁻¹𝐷𝑃₀₀, (63) where 𝐹=(𝑐−2,𝑐−3,…,1) (64) is a vector. Submitting (53) and (63) into (55), we can obtain 𝑃₀₀.

The matrices 𝐴⁻¹ and 𝐶⁻¹ can be computed iteratively. Let 𝑥_𝑖𝑗 denote the elements of matrix 𝐴⁻¹ and let 𝑦_𝑖𝑗 denote the elements of matrix 𝐶⁻¹. Then, we have 𝑥_𝑖𝑗𝑥=0,𝑖<𝑗,𝑗=2,3,…,𝑐−1,_𝑗𝑗=1𝑥𝑗𝜉,𝑗=1,2,…,𝑐−1,_𝑖𝑗=1𝑎𝑖𝜉^𝑖−1𝑥_{(𝑖−1)𝑗}−𝜆𝑥_{(𝑖−2)𝑗},𝑖>𝑗,𝑗=1,2,…,𝑐−1. (65) See the appendix for the proof of (65).

Since the matrix 𝐶 has the same structure as the matrix 𝐴, using (65), it is easy to get the elements of 𝐶⁻¹ as follows: 𝑦_𝑖𝑗𝑦=0,𝑖<𝑗,𝑗=2,3,…,𝑐−2,_𝑗𝑗=1𝜇𝑦𝑗+1,𝑗=1,2,…,𝑐−2,_𝑖𝑗=1()𝜇𝑏𝑖+1^𝑖𝑦_{(𝑖−1)𝑗}−𝜆𝑦_{(𝑖−2)𝑗},𝑖>𝑗,𝑗=1,2,…,𝑐−2. (66)

Using (65) and (66), it is easy to get from (62) that 𝑃₁₀=𝜆−𝜇𝛿𝑥₁₁𝑃₀₀,𝑃_𝑗0=𝜆−𝜇𝛿𝑥_𝑗1−𝜆𝑥_𝑗2𝑃₀₀,𝑗=2,3,…,𝑐−1,𝑃_(𝑗+1)1=𝑏₁𝑦_𝑗1−𝜆𝑦_𝑗2𝛿𝑃₀₀−𝛾^𝑗_𝑘=1𝑦_𝑗𝑘𝑃_𝑘0,𝑗=1,2,…,𝑐−2. (67)(68)(69)

Define 𝜙₀=𝑐−1+^𝑐−2_𝑗=1𝑐−𝑗−1𝑏₁𝑦_𝑗1+𝜆𝑦_𝑗2,𝜙_𝑘=^𝑐−2_𝑗=𝑘𝑐−𝑗−1𝑦_𝑗𝑘,𝑘=1,2,…,𝑐−2. (70)

Then, using (69), we have 𝑄(1)=^𝑐−1_𝑗=1𝑐−𝑗𝑃_𝑗1=Δ𝜙𝑃₀₀, (71) where Δ𝜙=𝛿𝜙₀−𝛾^𝑐−2_𝑘=1𝜙_𝑘𝜆−𝜇𝛿𝑥_𝑘1−𝜆𝑥_𝑘2. (72) Substituting (53) and (71) into (55), we get 𝑃₀₀=𝐾𝛾𝛾+𝜉𝜇−𝜆𝜉Δ𝜙,𝜇𝛾+𝜉𝜇−𝜆+𝐾𝛾𝜇𝜇+𝜉 (73) where 𝐾 is defined by (15).

Define 𝜓₀=𝑐−1+^𝑐−2_𝑗=1𝑗+1𝑐−𝑗−1𝑏₁𝑦_𝑗1+𝜆𝑦_𝑗2,𝜓_𝑘=^𝑐−2_𝑗=𝑘𝑗+1𝑐−𝑗−1𝑦_𝑗𝑘,𝑘=1,2,…,𝑐−2. (74) Then, using (69), we have 𝑄^′(1)=^𝑐−1_𝑗=1𝑗𝑐−𝑗𝑃_𝑗1=Δ𝜓𝑃₀₀, (75) where Δ𝜓=𝛿𝜓₀−𝛾^𝑐−2_𝑘=1𝜓_𝑘𝜆−𝜇𝛿𝑥_𝑘1−𝜆𝑥_𝑘2. (76)

3.5. Special Cases

In this subsection, we consider two special cases: 𝑐=1 and 𝑐=2. The performance measures of the system for these two cases are distinguished by using the superscripts M/M/1 and M/M/2, respectively.

Case 1 (single server model). If 𝑐=1, then 𝑄(1)=0 and 𝑄^′(1)=0. Thus, from (36) and (37), we have 𝑃^M/M/1_.0=1−𝜌𝛾+𝜉𝛾+𝜉1−𝜌,𝑃^M/M/1_.1=𝜌𝛾.𝛾+𝜉1−𝜌 (77) These results agree with the results given by Altman and Yechiali [12, Equation (2.17), page 264]. From (26) and (27), we get 𝐸[𝐿₀]^M/M/1=𝜆1−𝜌,𝛾+𝜉1−𝜌𝐸[𝐿₁]^M/M/1=𝜇−𝜆𝛾𝐸[𝐿₀(𝐿₀−1)]^M/M/1+2𝜇𝛾𝐸[𝐿₀]^M/M/1×2(𝜇−𝜆)²⁻¹. (78)(79) These results agree with the results given by Altman and Yechiali [12, pages 264-265].
From (39), we get 𝑃^M/M_𝑠^/1=𝛾.𝛾+𝜉1−𝜌 (80) This agrees with the result given by Altman and Yechiali [12, Equation (3.32), page 270].
From (31) and (40), we get 𝐸[𝐿₀(𝐿₀−1)]^M/M/1=2𝜆𝛾+2𝜉𝐸[𝐿₀]^M/M/1,𝑅^M/M_𝑎^/1=𝜉𝐸[𝐿₀]^M/M/1, (81) where 𝐸[𝐿₀]^M/M/1 is given by (78).

Case 2 (two-server model). If 𝑐=2, then 𝑄(1)=𝑄^′(1)=𝑃₁₁. Using (53) and substituting 𝑄(1)=𝑃₁₁ into (55), we obtain 𝑃^M/M/2₀₀=𝛾+𝜉1−𝜌𝜇1+𝛾𝛿.𝛾+𝜉1−𝜌+𝛿𝛾+𝜉 (82)
Thus, from (36) and (37), we get 𝑃^M/M/2_.0=1−𝜌𝛾+𝜉−𝛿𝛾+𝜉1−𝜌𝛾+𝜉𝑃𝛾+𝜉1−𝜌^M/M/200,𝑃^M/M/2_.1=𝜌𝛾+𝛿𝛾+𝜉1−𝜌𝛾+𝜉𝑃𝛾+𝜉1−𝜌^M/M/200, (83)(84) where 𝑃^M/M/2₀₀ is given by (82).
From (26) and (32), we get 𝐸[𝐿₀]^M/M/2=𝜆1−𝜌−𝛾+𝜉1−𝜌𝜆𝛿𝑃𝛾+𝜉1−𝜌^M/M/200,𝐸[𝐿₁]^M/M/2=𝜌𝛾1−𝜌+1𝛾+2𝜉𝜆1−𝜌𝐸[𝐿₀]^M/M/2+𝛿(1−𝜌)²𝑃^M/M/2₀₀, (85)(86) where 𝑃^M/M/2₀₀ is given by (82) and 𝐸[𝐿₀]^M/M/2 is given by (85).
Substituting (84) and 𝑄(1)=𝑃₁₁=𝛿𝑃₀₀ into (39), we get 𝑃^M/M_𝑠^/2=2𝛾+𝛿𝛾+𝜉1−𝜌𝛾+𝜉1+𝜌𝜌𝑃𝛾+𝜉1−𝜌^M/M/200, (87) where 𝑃^M/M/2₀₀ is given by (82).
From (31) and (40), we get 𝐸[𝐿₀(𝐿₀−1)]^M/M/2=2𝜆𝛾+2𝜉𝐸[𝐿₀]^M/M/2,𝑅_𝑎=𝜉𝐸[𝐿₀]^M/M/2, (88) where 𝐸[𝐿₀]^M/M/2 is given by (85).

4. Numerical Results

In this section, we investigate numerically the effects of the parameter 𝜉 and the number 𝑐 of servers on some performance measures.

For M/M/1 queueing model with impatient customers and multiple vacation policy, Altman and Yechiali [12] show that the probability 𝑃_.0[𝑃_.1] is an increasing (decreasing) concave (convex) function of 𝜉, having its limits at 1−(𝜆/𝜇) [at (𝜆/𝜇)]. Also, they show that 𝐸[𝐿₀] behaves similar to 𝑃_.1. In Figures 6 and 7, we consider whether the above property holds for our multiserver model, where the parameters are given as follows: 𝜆=1, 𝜇=2, and 𝛾=0.4.

Figure 6 

Effect of 𝜉 on 𝑃.0.

Figure 7 

Effect of 𝜉 on 𝑃.1.

From Figures 6 and 7, it is observed that 𝑃_.0 is an increasing function of 𝜉 for 𝑐=1 and 𝑃_.1 is a decreasing function of 𝜉 for 𝑐=1. However, this property may not hold for our multiserver model. This can be observed from the other three curves for 𝑐=10,20,30 in Figure 6 and Figure 7.

Altman and Yechiali [12] show that the probability 𝑃_.0 and 𝑃_.1 have their limits when 𝜉→0. We observe from Figures 6 and 7 that this property may also hold for our multiserver model. Further, we observe that their limits for the cases of 𝑐=5,10,20,30 when 𝜉→0 are very close.

From Figure 8, it is observed that 𝐸[𝐿₀] is a decreasing function of 𝜉 for both the case of single server 𝑐=1 and cases of multiserver 𝑐=5,10,20,30. This is because that the increasing of 𝜉 leads to the increasing of the number of reneging customers, which results in the decreasing of the mean system size 𝐸[𝐿₀]. Further, we observe that the four curves for 𝑐=5,10,20,30 are very close. This means that when 𝑐 is large enough 𝐸[𝐿₀] is insensitive to the number 𝑐 of servers.

Figure 8 

Effect of 𝜉 on 𝐸[𝐿0].

From Figure 9, we observe that 𝐸[𝐿₁] is a decreasing function of 𝜉 for the case of single server 𝑐=1. This can be verified analytically from (32) by letting 𝑐=1. However, 𝐸[𝐿₁] may not be a decreasing function of 𝜉 for the case of multiserver. This can be observed from the two curves of 𝑐=20 and 𝑐=30. Further, we observe that 𝐸[𝐿₁] varies significantly with the increasing of 𝑐. This means that it is sensitive to the number 𝑐 of servers.

Figure 9 

Effect of 𝜉 on 𝐸[𝐿1].

5. Conclusions

In this paper, we have studied an M/M/𝑐 queueing system with impatient customers and a synchronous vacation policy, where the customer impatience is due to the server being on vacation. We have derived some performance measures for the system in terms of two indexes 𝑄(1) and 𝑄^′(1). Based on these results, we have obtained some results about limiting behavior for some performance measures. In addition, we have obtained the iterative formulas for calculating these two indexes 𝑄(1) and 𝑄^′(1). Furthermore, we have derived closed-form expressions of some important performance measures for two special cases. The effects of the reneging rate and the number of servers on the some performance measures have been investigated numerically. The results obtained in this paper may have potential applications in production-inventory systems and communication networks.

Appendix

Proof of (65). Let 𝑋_𝑗=(𝑥_1𝑗,𝑥_2𝑗,…,𝑥_{(𝑐−1)𝑗})^𝑇, 𝑗=1,2,…,𝑐−1, be the 𝑗th column vector of the inverse matrix 𝐴⁻¹ and let 𝜀_𝑗=(0,…,1,…,0)^𝑇 be the 𝑗th unit column vector; then we have 𝐴𝑋_𝑗=𝜀_𝑗,𝑗=1,2,…,𝑐−1. (A.1) For 𝑗=1,2,…,𝑐−1, (A.1) can be rewritten as the following set of equations: 𝜆𝑥_𝑖−2𝑗−𝑎_𝑖−1𝑥_𝑖−1𝑗+𝑖𝜉𝑐_𝑖𝑗=0,𝑖≠𝑗,𝑖=1,2,…,𝑐−1,𝜆𝑥_𝑖−2𝑗−𝑎_𝑖−1𝑥_𝑖−1𝑗+𝑖𝜉𝑥_𝑖𝑗=1,𝑖=𝑗,𝑖=1,2,…,𝑐−1, (A.2)(A.3) where 𝑥_0𝑗 and 𝑥_−1𝑗 are defined to be zero. Repeating the use of (A.2) gives 𝑥_𝑖𝑗=0,𝑖=1,2,…,𝑗−1. (A.4)

Substituting (A.4) into (A.3) yields 𝑥_𝑗𝑗=1.𝑗𝜉 (A.5) From (A.2), we have 𝑥_𝑖𝑗=1𝑎𝑖𝜉^𝑖−1𝑥_𝑖−1𝑗−𝜆𝑥_𝑖−2𝑗,𝑖=𝑗+1,𝑗+2,…,𝑐−1. (A.6) This completes the proof.

Conflict of Interests

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

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 71071133), in part by the Natural Science Foundation of Hebei Province, China (no. G2012203136), and in part by MEXT, Japan.