Optimization of Balking and Reneging Queue with Vacation Interruption under 𝑁-Policy

Abstract

This paper analyzes a finite buffer multiple working vacations queue with balking, reneging, and vacation interruption under 𝑁-policy. In the working vacation, a customer is served at a lower rate and at the instants of a service completion; if there are at least 𝑁 customers in the queue, the vacation is interrupted and the server switches to regular busy period otherwise continues the vacation. Using Markov process and recursive technique, we derive the stationary system length distributions at arbitrary epoch. Various performance measures and some special models of the system are presented. Cost analysis is carried out using particle swarm optimization and quadratic fit search method. Finally, some numerical results showing the effect of model parameters on key performance measures of the system are presented.

1. Introduction

Performance modeling of queueing systems with balking and reneging has attracted many researchers owing to their wide applications in real life congestion problems such as impatient telephone switchboard customers, hospital emergency rooms handling critical patients and perishable goods storage in inventory systems. Balking and reneging are a common phenomena in queues; as a consequence, the customers either decide not to join the queue or depart after joining the queue without getting service due to impatience. Modeling balking and reneging is worthwhile because one obtains new managerial insights. The lost revenues due to balking and reneging in various industries can be enormous. While making decision for the number of servers needed in the service system to meet time-varying demand, the balking and reneging probabilities can be used to estimate the amount of lost business in more practical consideration for the managers as given in Liao [1].

Impatience is the most prominent characteristic as individuals always feel anxious and impatient during waiting for service in real life. The customer's impatient acts should be involved in the study of queueing system to model real situations exactly. Intermittent operation of a service can be economically appealing whenever full time service would result in significant server idle time or would preclude the use of the server in some other productive capacity. On the other hand, having the server inoperative for periods of time may increase the probability of customer losses due to balking and reneging. An 𝑀/𝑀/1 queue with customers balking and reneging has been discussed in Haight [2] and Haight [3], respectively. The combined impact of balking and reneging with finite capacity in an 𝑀/𝑀/1 queue has been studied by Ancker and Gafarian [4, 5]. Abou-El-Ata [6] discussed the finite buffer single server queueing system with balking and reneging. Analytical solutions of the single server Markovian overflow queue with balking, reneging and an additional server for longer queue were discussed in Abou-El-Ata and Shawky [7]. A computational algorithm and parameter optimization for a multiserver queue with unreliable server and impatient customers have been discussed by Chia and Jau-Chaun [8]. Choudhury and Medhi [9] presented some aspects of balking and reneging in finite buffer queue. Recently, Rakesh and Sumeet [10] studied an 𝑀/𝑀/1/𝑁 queueing model with retention of reneged customers and balking.

Queueing systems with server vacations have been studied due to their wide applications in several areas including computer and communication systems, manufacturing and production systems. Queueing systems with server vacations have been studied extensively in Tian and Zhang [11], Ke et al. [12], and so forth. Yue et al. [13] have analyzed an 𝑀/𝑀/1/𝑁 queue with balking, reneging, and server vacations. Vikas and Deepali [14] studied the state dependent bulk service queue with balking, reneging, and server vacation. However, there are numerous situations where the server remains active during the vacation period. Under working vacation (𝑊𝑉) policy, the server can provide service at a lower speed during vacation period rather than stopping service completely. At the end of a vacation if the queue is nonempty, a service period begins with normal service rate; otherwise the server takes another vacation. This policy is called multiple working vacations (𝑀𝑊𝑉𝑠) as introduced by Servi and Finn [15]. Recently, Vijaya Laxmi et al. [16] analyzed an 𝑀/𝑀/1/𝑁 working vacations queue with balking and reneging.

In order to utilize the server effectively, vacation interruption has become an important aspect, wherein the server interrupts the vacation once some indices of the system, such as the number of waiting customers, achieve a certain value in the vacation period. Li and Tian [17] considered a working vacation 𝑀/𝑀/1 queue with vacation interruptions. Baba [18] studied an 𝑀/𝑃𝐻/1 queue with working vacations and vacation interruptions. Zhang and Hou [19] studied the more general 𝑀/𝐺/1 queue with working vacations and vacation interruptions. An 𝑀/𝑀/1/𝑀𝑊𝑉 queue with 𝑁-policy has been studied in Zhang and Xu [20].

Queueing models with 𝑁-policy consider the most common issue of controlling arrivals and reducing the set up costs. The closed-form results in case of impatient customers with working vacations, vacation interruptions under 𝑁-policy are still not available. This motivated us to study an 𝑁-policy vacation interruptions queue with balking and reneging. This paper also discusses cost optimization problem under a given cost structure using particle swarm optimization (PSO) and quadratic fit search method (QFSM). PSO is a parallel evolutionary computation technique developed by Kennedy and Eberhart [21] based on the social behavior of animals such as bird flocking, fish schooling, and swarm theory. In the basic PSO algorithm the particle velocities build up too fast, and the optimum of the objective function is skipped. Hence, an inertia term introduced by Shi and Eberhart [22] is added to reduce the velocity. PSO can be easily implemented and is computationally inexpensive. An analysis of the PSO algorithm has been given in Qinghai [23] and Rao [24]. QFSM uses a 3-point pattern for fitting a quadratic function that has a unique optimum, see Rardin [25].

This paper deals with a finite buffer single server queue wherein customers may either balk or renege and the server takes multiple working vacations with a provision for vacation interruption. The server interrupts the vacation when the queue size reaches 𝑁; otherwise the vacation continues. The steady-state probabilities are obtained at arbitrary epoch through recursive technique which is easy to implement. We formulate a cost model to determine the optimum service rate during normal busy period and during working vacation period using PSO and QFSM.

The rest of the paper is organized as follows. Section 2 presents the description of the model and the steady-state probabilities. Sections 3 and 4 present some special cases, various performance measures, and cost analysis, respectively. Numerical results in the form of tables and graphs are presented in Section 5. Section 6 concludes the paper.

2. Model Description and Steady-State Probabilities

We consider a multiple working vacations 𝑀/𝑀/1/𝐾 queueing system with vacation interruptions wherein customers either balk or renege. The following notations and assumptions have been used throughout the paper. (i)Customers arrive one at a time according to Poisson process with arrival rate 𝜆. On arrival a customer either decides to join the queue or balk. Let 𝑏_𝑖 represent the probability with which the customer either decides to join the queue or balk with probability 1−𝑏_𝑖, when there are 𝑖 customers ahead of him in the system. Furthermore, we assume that 0≤𝑏_𝑖+1≤𝑏_𝑖<1,1≤𝑖≤𝐾−1, 𝑏₀=1, and 𝑏_𝐾=0. (ii)After joining the queue each customer will wait a certain length of time, say 𝑇, for service to begin before they get impatient and leave the queue without receiving service. This time 𝑇 is assumed to follow exponential distribution with mean 1/𝛼. Since the arrival and departure of an impatient customer without receiving service are independent, the average rate of reneging of a customer is given by (𝑖−1)𝛼. Hence, the function of customer's average reneging rate is given by 𝑟_𝑖=(𝑖−1)𝛼, 1≤𝑖≤𝐾; 𝑟_𝑖=0,𝑖>𝐾.(iii)The customers are served on a first-come first-served (FCFS) queue discipline. The service times in normal busy period are assumed to follow exponential distribution with mean 1/𝜇. (iv)The server takes working vacations whenever the system becomes empty. At a service completion epoch during vacation, if there are at least 𝑁 customers waiting in the system, the server ends the vacation and switches to normal service; that is, a vacation interruption occurs; otherwise, it continues the vacation. Furthermore, at vacation termination epoch, if there are less than 𝑁 customers in the queue, the server begins another vacation; else, it switches to the normal busy period. (v)During any working vacation the server will serve customers at a rate different (generally lower) from the normal service rate 𝜇. The vacation duration and the service times in vacation period are also assumed be exponentially distributed with mean 1/𝜙 and 1/𝜂, respectively. (vi)The interarrival times, vacation times, service times during normal busy period, and service times during vacation are mutually independent.

2.1. Steady-State Equations

At steady-state, let 𝑃_0,𝑖, 0≤𝑖≤𝐾, be the probability that there are 𝑖 customers in the system when the server is in working vacation and let 𝑃_1,𝑖, 1≤𝑖≤𝐾, be the probability that there are 𝑖 customers in the system when the server is in normal busy period. Using the Markov theory, the following set of steady-state equations may be written 𝜆𝑃_0,0=𝜂𝑃_0,1+𝜇𝑃_1,1,𝜆𝑏_𝑖+𝜂+(𝑖−1)𝛼𝑃_0,𝑖=𝜂+𝑖𝛼𝑃_0,𝑖+1+𝜆𝑏_𝑖−1𝑃_0,𝑖−1,1≤𝑖≤𝑁−1,𝜆𝑏_𝑖+𝜂+(𝑖−1)𝛼+𝜙𝑃_0,𝑖=𝜆𝑏_𝑖−1𝑃_0,𝑖−1+𝑖𝛼𝑃_0,𝑖+1,𝑁≤𝑖≤𝐾−1,𝜂+(𝐾−1)𝛼+𝜙𝑃_0,𝐾=𝜆𝑏_𝐾−1𝑃_0,𝐾−1,𝜇+𝜆𝑏_𝑖+(𝑖−1)𝛼𝑃_1,𝑖=𝜆𝑏_𝑖−1𝑃_1,𝑖−1+𝜇+𝑖𝛼𝑃_1,𝑖+1,1≤𝑖≤𝑁−1,𝜇+𝜆𝑏_𝑖+(𝑖−1)𝛼𝑃_1,𝑖=𝜆𝑏_𝑖−1𝑃_1,𝑖−1+𝜇+𝑖𝛼𝑃_1,𝑖+1+𝜂𝑃_0,𝑖+1+𝜙𝑃_0,𝑖,𝑁≤𝑖≤𝐾−1,𝜇+(𝐾−1)𝛼𝑃_1,𝐾=𝜆𝑏_𝐾−1𝑃_1,𝐾−1+𝜙𝑃_0,𝐾. (1)(2)(3)(4)(5)(6)(7) The steady-state probabilities 𝑃_0,𝑖(𝑃_1,𝑖) can be obtained by solving the previous system of equations recursively that is presented as a theorem later.

Theorem 1. The steady-state probabilities are given by 𝑃_0,𝑖=𝜁_𝑖∑^𝐾_𝑚=0𝜁_𝑚+∑^𝐾_𝑚=1𝜉_𝑚,0≤𝑖≤𝐾,𝑃_1,𝑖=𝜉_𝑖∑^𝐾_𝑚=0𝜁_𝑚+∑^𝐾_𝑚=1𝜉_𝑚,1≤𝑖≤𝐾, (8)(9) where 𝜁_𝐾=1,𝜁_𝐾−1=𝜙+𝜂+(𝐾−1)𝛼𝜆𝑏_𝐾−1,𝜁_𝑖=𝜆𝑏_𝑖+1+𝜂+𝜙+𝑖𝛼𝜁_𝑖+1−(𝑖+1)𝛼𝜁_𝑖+2𝜆𝑏_𝑖,𝜁𝑖=𝐾−2,𝐾−3,…,𝑁−1,_𝑖=𝜆𝑏_𝑖+1+𝜂+𝑖𝛼𝜁_𝑖+1−𝜂+(𝑖+1)𝛼𝜁_𝑖+2𝜆𝑏_𝑖,𝜉𝑖=𝑁−2,𝑁−3,…,0,₀=0,𝜉₁=𝜆𝜁₀−𝜂𝜁₁𝜇,𝜉_𝑖=𝜇+𝜆𝑏_𝑖−1+(𝑖−2)𝛼𝜉_𝑖−1−𝜆𝑏_𝑖−2𝜉_𝑖−2()𝛼,𝜉𝜇+𝑖−1𝑖=2,3,…,𝑁,_𝑖=𝜇+𝜆𝑏_𝑖−1+(𝑖−2)𝛼𝜉_𝑖−1−𝜆𝑏_𝑖−2𝜉_𝑖−2−𝜂𝜁_𝑖−𝜙𝜁_𝑖−1()𝛼,𝜇+𝑖−1𝑖=𝑁+1,𝑁+2,…,𝐾. (10)

Proof. Solving the system of (1) to (6) recursively and using the normalization condition ∑^𝐾_𝑖=0𝑃_0,𝑖+∑^𝐾_𝑖=1𝑃_1,𝑖=1, we obtain the results of the theorem.

3. Special Cases

In this section, results pertaining to some models have been deduced from our model by taking specific values for the parameters 𝛼, 𝜂, 𝑏_𝑖, and 𝑁.

Case 1 (𝛼→0, 𝑏_𝑖=1, 0≤𝑖≤𝐾, 𝑁=1). The model reduces to 𝑀/𝑀/1/𝐾 queue with multiple working vacations and vacation interruption. We can obtain the steady-state probabilities 𝑃_0,𝑖(0≤𝑖≤𝐾) and 𝑃_1,𝑖(1≤𝑖≤𝐾) from (8)-(9), where 𝜁_𝐾=1,𝜁_𝐾−1=𝜙+𝜂𝜆,𝜁_𝑖=𝜆+𝜂+𝜙𝜆𝜁_𝑖+1,𝜉𝑖=𝐾−2,𝐾−3,…,0,₀=0,𝜉₁=𝜆𝜁₀−𝜂𝜁₁𝜇,𝜉_𝑖=𝜇+𝜆𝜉_𝑖−1−𝜆𝜉_𝑖−2−𝜂𝜁_𝑖−𝜙𝜁_𝑖−1𝜇,𝑖=2,3,…,𝐾, (11) which match with the results of 𝑀/𝑀/1/∞ queue with multiple working vacations and vacation interruption Li and Tian [17] for large 𝐾 values.

Case 2 (𝑁=1 and without vacation interruptions). The model reduces to 𝑀/𝑀/1/𝐾 queue with balking, reneging and multiple working vacations and results match numerically with the results available in Vijaya Laxmi et al. [16].

Case 3 (𝑁=1, 𝜂→0). The model reduces to 𝑀/𝑀/1/𝐾 queue with balking, reneging and multiple vacations. Taking the average reneging rate as 𝑟_𝑖=(𝑖−𝑛)𝛼, 𝑛=0 in vacation period and 𝑛=1 in normal busy period, we can obtain the steady-state probabilities 𝑃_0,𝑖 and 𝑃_1,𝑖 from (8)-(9), where 𝜁_𝐾=1,𝜁_𝐾−1=𝐾𝛼+𝜙𝜆𝑏_𝐾−1,𝜉₀=0,𝜉₁=𝜆𝜁₀−𝛼𝜁₁𝜇,𝜁_𝑖=𝜆𝑏_𝑖+1+𝜙+(𝑖+1)𝛼𝜁_𝑖+1−(𝑖+2)𝛼𝜁_𝑖+2𝜆𝑏_𝑖,𝜉𝑖=𝐾−2,𝐾−3,…,0,_𝑖=𝜇+𝜆𝑏_𝑖−1+(𝑖−2)𝛼𝜉_𝑖−1−𝜆𝑏_𝑖−2𝜉_𝑖−2−𝜙𝜁_𝑖−1()𝛼,𝜇+𝑖−1𝑖=2,3,…,𝐾. (12) The previous results match numerically with the results of Yue et al. [13].

Case 4 (𝜂→0 and 𝜙→∞). The model reduces to 𝑀/𝑀/1/𝐾 queue with balking and reneging. These results match with the results available in the literature.

4. Performance Measures

Once the steady-state probabilities are obtained, one can evaluate various performance measures of the model like probability that the server is in normal busy period (𝑃_𝑏), probability that the server is in working vacation (𝑃_𝑤𝑣), average queue length (𝐿_𝑞), average system length (𝐿_𝑠), and they are given by 𝑃_𝑏=^𝐾_𝑖=1𝑃_1,𝑖,𝑃_𝑤𝑣=^𝐾_𝑖=0𝑃_0,𝑖,𝐿_𝑞=^𝐾_𝑖=1(𝑖−1)𝑃_0,𝑖+𝑃_1,𝑖,𝐿_𝑠=^𝐾_𝑖=1𝑖𝑃_0,𝑖+𝑃_1,𝑖. (13) When a customer finds 𝑖 customers in the system upon his arrival, then the instantaneous balking rate is 𝜆(1−𝑏_𝑖), as the probability that a customer balks in the system is 1−𝑏_𝑖. Using the concepts of Ancker and Gafarian [4, 5], we can obtain the average balking rate (𝐵.𝑅.) as 𝐵.𝑅.=^𝐾_𝑖=1𝜆1−𝑏_𝑖𝑃_0,𝑖+^𝐾_𝑖=1𝜆1−𝑏_𝑖𝑃_1,𝑖. (14) If there are 𝑖 customers in the system and server is available, then there are (𝑖−1) waiting customers in the queue. Since any one of the (𝑖−1) customers in the queue may renege, the instantaneous reneging rate is (𝑖−1)𝛼. Again, following the models of Ancker and Gafarian [4, 5], the average reneging rate (𝑅.𝑅.) is given by 𝑅.𝑅.=^𝐾_𝑖=1(𝑖−1)𝛼𝑃_0,𝑖+^𝐾_𝑖=1(𝑖−1)𝛼𝑃_1,𝑖. (15) The average rate of customer loss (𝐿.𝑅.) is the sum of the average balking rate and the average reneging rate. Thus, we have 𝐿.𝑅.=𝐵.𝑅.+𝑅.𝑅. (16)

4.1. Cost Model

In practice, queueing managers are always interested in minimizing operating cost of unit time. In this section, we first develop a steady-state expected cost function per unit time, in which either service rate during normal busy period or service rate during working vacation period is the decision variable. Our objective is to determine the optimum value of 𝜇 or 𝜂 to minimize the cost function.

Let us define the following cost elements: 𝐶_𝜇≡ service cost per unit time during normal busy period, 𝐶_𝜂≡ service cost per unit time during working vacation period, 𝐶_𝑙𝑞≡ cost per unit time when a customer joins the queue and waits for service, 𝐶_𝑙𝑟≡ cost per unit time when a customer balks or reneges.

The cost parameters are assumed to be linear in the average number of the indicated quantities such as the average queue length (𝐿_𝑞) and the average rate of customer loss (𝐿.𝑅.). Based on the definitions of each cost element listed previously and its corresponding system performance measures, the cost minimization problem can be illustrated mathematically as: Minimize∶𝑓(𝑥)=𝐶_𝜇𝜇+𝐶_𝜂𝜂+𝐶_𝑙𝑞𝐿_𝑞+𝐶_𝑙𝑟𝐿.𝑅., (17) where 𝑥 can be either service rate during normal busy period (𝜇) or service rate during working vacation period (𝜂). The objective is to determine the optimal service rate 𝜇^∗ or 𝜂^∗ to minimize the cost function 𝑓. It is a difficult task to develop analytic results for the optimum value of 𝜇 or 𝜂, because the expected cost function is highly complex. We note that the derivatives of the operating cost function per unit time are not easily available. We solve the above stated optimization problem using PSO and QFSM. Both methods give identical results as shown in the next section.

4.1.1. Algorithm for PSO

Step 1. Initialize a population of particles of size 𝑚 with random positions in the problem space. For each particle, evaluate the objective function value.

Step 2. As all the particles will be moving to the optimal point with a velocity, find the velocities of particles. Initially, all particle velocities are assumed to be zero.

Step 3. Store each particle's coordinates associated with the best solution in 𝑝best and the coordinates of the particle associated with the overall best solution in 𝑔best.

Step 4. Compare each particle's fitness with particle's 𝑝best. If current value is better, then set 𝑝best value to the current value. Compare each particle's fitness with the population's overall previous best (𝑔best), and if any particle's current value is better than 𝑔best, then set 𝑔best to the current value.

Step 5. Change the velocity 𝑣_𝑗(𝑖) and position 𝑥_𝑗(𝑖) of the 𝑗th particle in the 𝑖th iteration according to the following equations, respectively, 𝑣_𝑗(𝑖)=𝜑(𝑖)𝑣_𝑗(𝑖−1)+𝑐₁rand(⋅)𝑝best𝑗−𝑥_𝑗(𝑖−1)+𝑐₂rand(⋅)𝑔best−𝑥_𝑗()𝑥𝑖−1,1≤𝑗≤𝑚,_𝑗(𝑖)=𝑥_𝑗(𝑖−1)+𝑣_𝑗(𝑖),1≤𝑗≤𝑚, (18) where rand(⋅) is a random number between (0,1), 𝑐₁ and 𝑐₂ are learning factors, and 𝜑(𝑖) is the inertia weight given by 𝜑(𝑖)=𝜑_max−𝜑_max−𝜑_min𝑖_max𝑖 (19) with 𝜑_max, 𝜑_min being the maximum and minimum values of the inertia weight (𝜑_max=0.9, 𝜑_min=0.4 are commonly used) and 𝑖_max the maximum number of iterations.

Step 6. Loop to Step 3 until the stopping criterion is met. The stopping criterion is that the number of iterations is completed or the difference between two successive 𝑔best values is less than a predefined error tolerance.

4.1.2. Algorithm for QFSM

Given a 3-point pattern, we can fit a quadratic function through corresponding functional values that have a unique minimum, 𝑥^𝑞, for the given objective function 𝑓(𝑥). Quadratic fit uses this approximation to improve the current 3-point pattern by replacing one of its points with approximate optimum 𝑥^𝑞. The unique optimum 𝑥^𝑞 of the quadratic function agreeing with 𝑓(𝑥) at 3-point operation (𝑥^𝑙,𝑥^𝑚,𝑥^ℎ) given by Rardin [25] occurs at 𝑥^𝑞≅12𝑓𝑥^𝑙𝑠^𝑚−𝑠^ℎ+𝑓𝑥^𝑚𝑠^ℎ−𝑠^𝑙+𝑓𝑥^ℎ𝑠^𝑙−𝑠^𝑚×𝑓𝑥^𝑙𝑥^𝑚−𝑥^ℎ+𝑓𝑥^𝑚𝑥^ℎ−𝑥^𝑙+𝑓𝑥^ℎ𝑥^𝑙−𝑥^𝑚−1, (20) where 𝑠^𝑙=(𝑥^𝑙)², 𝑠^𝑚=(𝑥^𝑚)², 𝑠^ℎ=(𝑥^ℎ)².

5. Numerical Discussions

In this section, we illustrate the results obtained in the previous sections numerically and discuss the effect of system parameters on the performance indices. The differences of parameters, such as the lower service rate (𝜂), vacation rate (𝜙), vacation interruption, and 𝑁 policy, also influence the various performance measures of the model. We fix the capacity of the system as 𝐾=10 and 𝑁=4, and the parameters of the model are assumed as 𝜆=1.8, 𝜇=2.5, 𝜂=1.9, 𝜙=1.2, and 𝛼=0.3, unless they are considered as variables. The balking function is taken as 𝑏_𝑖=1−(𝑖/𝐾²), 1≤𝑖≤𝐾−1, 𝑏₀=1, 𝑏_𝐾=0, and the various cost elements are chosen as 𝐶_𝜇=18, 𝐶_𝜂=15, 𝐶_𝑙𝑞=25, 𝐶_𝑙𝑟=10. The swarm size, maximum number of iterations, and learning factors of PSO algorithm are taken as 20, 50 and 𝑐₁=𝑐₂=2, respectively, and the tolerance of QFSM is 𝜅=1×10⁻⁶.

Table 1 presents the optimum values of 𝜇 along with 𝐿^∗_𝑞 and 𝐿.𝑅.^∗ for various 𝜂 and 𝜆, using particle swarm optimization. For a fixed 𝜆, decreasing trend is observed in 𝜇^∗, 𝐿^∗_𝑞, and 𝐿.𝑅.^∗ with the increase of 𝜂. For a fixed 𝜂, increasing trend is observed in 𝜇^∗, 𝐿^∗_𝑞, and 𝐿.𝑅.^∗ with the increase of 𝜆. But the minimum cost increases with the increase of 𝜂 and 𝜆.

Table 1 

Optimum values for various values of 𝜂 and 𝜆.

To ensure the reliability of PSO algorithm, we compared our results with QFSM, and the results are shown in Table 2. Though both these methods give identical results, the advantage of using PSO algorithm lies in the ease with which it can be tuned and implemented using only a velocity operator to drive the search throughout the search space, whereas the QFSM is dependent on the proper choice of the initial 3-point approximation. It is also observed that convergence is faster in PSO algorithm.

Table 2 

𝜂^∗ from PSO and QFSM.

The effect of 𝜇 and 𝜂 on various performance measures of the model, for various balking functions, 𝛼 and 𝜙, is presented in Tables 3 and 4, respectively. Note that for any 𝜇(𝜂), the balking function 𝑏_𝑖=1−(𝑖/𝐾²) yields the lowest 𝐿.𝑅. which supports our choice of the balking function as 𝑏_𝑖=1−(𝑖/𝐾²). On the other hand, for any choice of the balking function 𝑃_𝑏, 𝐿_𝑞, and 𝐿.𝑅. decrease with the increase of 𝜇(𝜂), while 𝑃_𝑤𝑣 and cost increase with 𝜇(𝜂), as intuitively expected. From the tables it can also be observed that for fixed 𝜇(𝜂), increase in 𝛼 results in the decrease of 𝑃_𝑏, 𝐿_𝑞, and cost, whereas the other performance measures 𝑃_𝑤𝑣 and 𝐿.𝑅. increase with 𝛼. Further, for fixed 𝛼, 𝑃_𝑏, 𝐿_𝑞, and 𝐿.𝑅. decrease with the increase of 𝜇(𝜂). It can also be observed that as 𝜇(𝜂) increases, 𝑃_𝑏, 𝐿_𝑞, and 𝐿.𝑅. decrease for any 𝜙. Moreover, for fixed 𝜇(𝜂), except 𝑃_𝑏 all the other performance measures decrease with the increase of 𝜙. This is because as 𝜙 increases, the vacation duration decreases and the server switches to normal busy period during which service is provided with rate 𝜇.

Table 3 

Various performance measures for different balking functions, 𝜇, 𝛼, and 𝜙.

Table 4 

Various performance measures for different balking functions, 𝜂, 𝛼, and 𝜙.

Figure 1 shows the effect of the arrival rate (𝜆) on the expected number of customers in the queue (𝐿_𝑞) with and without vacation interruptions (𝑉𝐼). It can be observed that with the increase of 𝜆, the expected queue length increases as it should be. Further, one may also observe that for higher values of 𝜆, 𝐿_𝑞 is less in the vacation interruptions model as compared to the model without vacation interruptions. This is due to the fact that in vacation interruptions model, the server tends to interrupt the vacation and switches to normal service rate when more customers arrive during vacation. The impact of arrival rate (𝜆) on the expected number of customers in the queue (𝐿_𝑞) with and without balking and reneging (𝐵&𝑅) is shown in Figure 2. It can be seen that as arrival rate increases 𝐿_𝑞 monotonically increases. Moreover, the model with balking and reneging outperforms the model without balking and reneging.

Figure 1 

Impact of 𝜆 on 𝐿𝑞 with and without vacation interruption.

Figure 2 

𝜆versus 𝐿𝑞 with and without balking and reneging.

Figures 3 and 4 plot the effect of the average rate of customer loss (𝐿.𝑅.) for different capacity of the system (𝐾) and threshold value (𝑁), respectively, with and without vacation interruptions. From the figures it can be seen that as the buffer size (𝐾) increases the customer loss decreases, whereas it increases with the increase of 𝑁, as intuitively expected. Further, in both the figures 𝐿.𝑅. is higher in the case of without vacation interruptions as compared to vacation interruptions model implying that a job has to wait more in models without vacation interruptions.

Figure 3 

Effect of 𝐾 on 𝐿.𝑅.

Figure 4 

Effect of 𝑁 on 𝐿.𝑅.

Figure 5 presents the effect of service rate during vacation (𝜂) on the average number of customers in the queue (𝐿_𝑞) for various vacation rates (𝜙). As expected, 𝐿_𝑞 monotonically decreases as 𝜂 increases for any 𝜙. Further, we observe that for 𝜂≤𝜇, the expected queue lengths decrease with the increase of 𝜙, and when 𝜂 crosses 𝜇, this trend is reversed. This shows that for better performance of the model 𝜂 should be chosen less than 𝜇. Figure 6 depicts the minimum and average costs for number of iterations considered in PSO, when 𝜂 is a decision variable. From the figure, it is observed that though initially the average cost varies widely from iteration to iteration, stability is attained after some iterations.

Figure 5 

Impact of 𝜂 on 𝐿𝑞.

Figure 6 

Iterations in PSO versus cost.

6. Conclusions

In this paper, we have carried out an analysis of a finite-buffer balking and reneging Markovian multiple working vacations queue with vacation interruptions under 𝑁-policy that have potential applications in modeling computer and telecommunication systems, computer networks, and so forth. We have obtained the closed-form expressions for the steady-state probabilities using recursive technique. Various performance measures such as expected number of customers in the system, average balking rate, average reneging rate, and the average rate of customer loss due to impatience are evaluated. We have also considered a cost optimization problem using particle swarm optimization and quadratic fit search method. Numerical results have been discussed which may be helpful to explore the effect of parameters on performance measures. The technique used in this paper can be applied to analyze more complex models such as 𝑀𝐴𝑃/𝑀/1/𝑁 and 𝑀/𝑀/𝑐/𝑁 multiple working vacations queues with balking and reneging which are left for future investigations.