Advances in Operations Research

Volume 2016, Article ID 4024950, 17 pages

http://dx.doi.org/10.1155/2016/4024950

## Phase-Type Arrivals and Impatient Customers in Multiserver Queue with Multiple Working Vacations

Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati 781 039, India

Received 27 October 2015; Accepted 14 February 2016

Academic Editor: Viliam Makis

Copyright © 2016 Cosmika Goswami and N. Selvaraju. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a PH/M/c queue with multiple working vacations where the customers waiting in queue for service are impatient. The working vacation policy is the one in which the servers serve at a lower rate during the vacation period rather than completely ceasing the service. Customer’s impatience is due to its arrival during the period where all the servers are in working vacations and the arriving customer has to join the queue. We formulate the system as a nonhomogeneous quasi-birth-death process and use finite truncation method to find the stationary probability vector. Various performance measures like the average number of busy servers in the system during a vacation as well as during a nonvacation period, server availability, blocking probability, and average number of lost customers are given. Numerical examples are provided to illustrate the effects of various parameters and interarrival distributions on system performance.

#### 1. Introduction

In communication networks, multiple servers are used to reduce the traffic congestion and improve the system performance. Multiple services are also used in highly efficient bandwidth-intensive applications. Different services may require different channel capacities and capacity of a channel depends upon the number of resources allocated to it. To understand the network behavior and to make intelligent decisions in their management, these systems can be modelled as multiserver queueing systems with server vacations. Levy and Yechiali [1] first discussed an M/M/c queue with exponentially distributed vacations. Tian and Li [2], Tian et al. [3], and Tian and Zhang [4] studied a variety of vacation models with multiple servers. They established the conditional stochastic decomposition properties on the steady-state queue length and the waiting time when all the servers are busy and obtained the stationary distributions for queue length and waiting times. Tian and Zhang [5] considered a two-threshold vacation policy in the context of a multiserver queueing model M/M/c. A multiserver queueing system with identical unreliable servers with PH-distributed service times is considered by Yang and Alfa [6]. Chakravarthy [7] studied an MAP/M/c queueing system, in which a group of servers take a simultaneous PH vacation.

The phenomenon of customer impatience is commonly observed in queueing systems, where customers leave a service system before receiving service due to the long waiting time or due to uncertainty of receiving service. Customer impatience or reneging represents loss in revenues and customer goodwill to the service provider. The problem of queues with impatient customers was first analyzed by Palm [8]. A bibliography can be found in Gross et al. [9]. Perel and Yechiali [10] considered a two-phase service impatient model where the customers become impatient if the server is in slow service phase. There are situations where customer’s impatience is due to the absence of the server, more precisely due to the server being on vacation, and is independent of the customers in system. Altman and Yechiali [11, 12] studied the customer impatience in a classical vacation model and system with additional task, respectively. Economou and Kapodistria [13] considered an unreliable queue where the customers leave the system at system failure times.

Multiserver queues with impatience, however, have attracted much attention in queueing literature possibly because of explosive demands to efficiently design and manage call or contact centres. Baccelli et al. [14] studied the waiting time distribution in M/M/c queue with general impatience bound on queueing times by constructing a simple Markov process and also gave the waiting time distribution in the M/G/1 queue with general impatience on queueing times. Yechiali [15] considered an M/M/c system which as a whole suffers occasionally a disastrous breakdown, upon which all present customers (waiting and served) are cleared from the system and lost. Stationary distribution of a multiserver vacation queue with constant impatient times is studied by Sakuma and Inoie [16]. Chen et al. [17] studied M/M/m/k queue with preemptive resume and impatience of the prioritised customers and derived the queue length distraction in stationary state and performance measures using the method of matrix analysis.

In communication systems, wavelength division multiplexing (WDM) is a method of transmitting packets from different sources, over the same fiber optic link, to the destination. A WDM network divides the available fiber bandwidth into WDM channels, details in Ho and Woei [18] and also in Wang [19]. This division of bandwidth or channel allocation is based on the capacities required for various services. For a high performance system, WDM channel allocation should lead to optimized resource utilization in a given network, which is physically feasible and cost-effective. A reconfigurable WDM system can be modelled as a queue with working vacations (WVs) as explained in Goswami and Selvaraju [20]. This vacation cannot be put in a classical vacation framework because here, unless the system is empty, the service does not cease completely. Servi and Finn [21] were the first to model such a WDM network into a WV queueing model. Liu et al. [22] studied the M/M/1/WV model with multiple WVs whereas the single WV model is analyzed by Tian et al. [23]. The same model is studied by Xu et al. [24] and also by Xiu et al. [25] with single WV and setup times. Wang et al. [26] presented the M/M/1/WV model using Newton’s method to compute the steady-state probabilities and system performance measures. Wu and Takagi [27] extended Servi and Finn’s work to M/G/1/WV model with generally distributed service times and vacation duration times. Baba [28] considered the GI/M/1/WV system with general independent arrival process where the distributions of the vacation duration times and service times are exponential. Chen et al. [29, 30] proposed an N-policy WV and a cyclic polling system for WDM taking the service times as exponential and PH distribution, respectively. Lin and Ke [31] considered a multiserver M/M/c queue and a cost model is derived to determine the optimal values of the number of servers and the WV rate simultaneously, in order to minimize the total expected cost per unit time.

Short distance networks, like local area networks (LANs), mostly use multimode WDM links. Multimode link is a single fiber link that supports many propagation paths or transverse modes through it. Aronson et al. [32] explained how the bandwidth of the fiber is multiplied by the number of paths used by using WDM in multimode fiber. LAN over Internet Protocol (IP) allows the forwarding of LAN packets over the Internet or an intranet network. One of the most critical performance measures in LAN over IP is the percentage of packets that are transmitted within hard delay bound or time constraint. If quality of service requirements is not met within the time bound, end users may terminate the Internet connections. A connection is terminated by pressing the stop button, refreshing the connection, or following a different link. This behavior can be termed the impatience of a user in LANs. To study the effect of multiple servers and user impatience on the performance in a WDM network, we consider in this paper a multiserver model with asynchronous multiple working vacation (AMWV) policy and impatient customers. In an AMWV policy, the servers take vacations individually and continue taking vacations till they do not find any customer in the system. An M/M/1/WV impatient model with single and multiple WV policies is studied by Selvaraju and Goswami [33]. Analysis of a finite buffer M/M/2 working vacations queue with balking and reneging wherein the servers operate under a triadic () policy is done recently by Laxmi and Jyothsna [34]. Lin and Ke [31] presented a multiserver WV queue with exponential interarrivals but none of these models represent systems with nonexponential arrivals or state-dependent systems. To study the role of arrival processes in a multiserver model having impatient customers, we consider here the PH arrival process. PH distribution is a general, nonexponential distribution characterized by a Markov chain. Importance of considering PH interarrivals is the fact that PH distribution is able to capture the nonexponential effects on arrivals while information flows in modern communication systems are rarely exponential. PH distribution is able to capture the profound effect of arrivals in system performance measures and makes the mathematical model more convincing to fit a real world scenario.

The paper is organized as follows. In Section 2, we formulate the system as a three-dimensional continuous-time Markov chain whose generator matrix is a level-dependent quasi-birth-death (QBD) process. Section 3 gives the finite truncation method used to find the stationary probability vector of the level-dependent process. The various performance measures are listed in Section 4 and in Section 5 the numerical illustrations of the system are presented.

#### 2. Model Description

We consider a PH/M/c queue with multiple WVs and impatient customers. The interarrival times of customers follow a PH distribution, PH, of dimension and with arrival rate . A PH distribution denotes the distribution of time until absorption in a finite Markov chain whose transition rate matrix is of type and is the initial probability vector satisfying and , where is the column vector of dimension with all the entries equal to one. The matrix is a nonsingular square matrix with , , and , . The matrix is a nonnegative, -dimensional column vector, grouping the absorption rates from any state to the absorbing one. The matrix gives the transition from one phase to another with an arrival of a customer to the system.

The customers are served according to FCFS basis. An arriving customer who finds all the servers busy has to wait in queue; that is, when the number of customers in the system is more than , a queue begins to form. The servers work independently of each other. The service times of each server during the nonvacation period follow an exponential distribution with rate , denoted by . A server goes to a WV as soon as it completes a service and finds no customer to serve in the system. For each server, the duration of WVs follows distribution. During a WV period of a server, if a customer arrives to that server, it will serve the customer with distribution, where ; that is, the customer will be served at a lower service rate. When a server returns from its vacation, if it finds at least one customer in queue waiting for service or finds an ongoing service in that server, the server switches its service rate from to and a nonvacation period starts. Otherwise, if the server finds an empty queue, after returning from one vacation, it immediately leaves for another WV.

An arriving customer gets service immediately upon its arrival, if it finds any of the servers empty. But if all the servers are busy, the customer has to wait in a queue. A waiting customer becomes impatient when it finds all the servers serving at rate ; that is, if the waiting customer finds all the servers in their WV period, the customer activates an impatient timer . This impatient timer follows distribution and is independent of the number of customers in the queue at that moment. If no server returns from its WV period by the time expires, the customer leaves the system and never returns. Otherwise, if any of the servers returns from its vacation before the time expires, the customer stops the timer and stays in the system until its service is completed. Here, the customer’s impatience depends not only on waiting time in a queue but also on the number of servers that are in WVs. The interarrival times, service times, vacation duration times, and the impatient times all are taken to be mutually independent.

To model this system, we define a continuous-time Markov chain:where denotes the total number of customers in the system, denotes the number of busy servers in nonvacation state, and gives the phase of the arrival process. The state space of this Markov chain isThe lexicographical order of the states, that is, , , , , , , , , gives the infinitesimal generator matrix of the Markov chain as withwhere, for , and, for , For , The matrix is an identity matrix of dimension . Here, for , dimension of the matrices , , and increases with the levels; and for , the matrices are of dimensions each. It can be observed that given above is the generator of a nonhomogeneous QBD process, which we assume to be irreducible, with levels denoting the number of customers in the system.

#### 3. Stationary Distribution

The queueing system under study is stable for [28, 35].

Let be the stationary probability vector associated with satisfyingAggregating terms depending on levels, we get . Further depending on number of busy servers in nonvacation, we get, for , , which are row -vectors and, for , are row -vectors. Each vector is an -dimensional row vector, for , depending on the phases of arrivals.

In this model, the generator matrix is spatially nonhomogeneous and a closed-form analytical solution or a direct algorithmic computation of the stationary probability vector is quite difficult, if not impossible. For such level-dependent QBDs (LDQBDs), the stationary vectors are usually approximated by using various numerical approximation methods like finite truncation method (Artalejo et al. [36] and Chakravarthy et al. [37]), generalized truncation method (Falin [38] and Artalejo and Pozo [39]), truncation method using LDQBD processes (Bright and Taylor [40] and Krishnamoorthy et al. [41]), and matrix-geometric approximations (Neuts and Rao [42]).

Different methods have different levels of computable efficiency but it is expected that whichever method is used, the general behavior of the performance measures of a system with a change in system parameters is not affected by the method used. Since the finite truncation method is comparatively tractable compared to the others, we choose this method to derive the stationary distributions of the nonhomogeneous QBD with the generator matrix given by (4).

In the finite truncation method, the infinite generator matrix is truncated at a finite level . That is, the system of equations given by and is truncated at a sufficiently large value, say , and the resulting finite system is solved for the equilibrium probability vector. The level is arbitrary but fixed and it is chosen such that customer loss probability due to truncation is small. As for higher dimension generator matrices, the level is difficult to find analytically; a trial-and-error approach needs to be adopted. An appropriate level, say , is determined by starting with a reasonable initial value for and increasing it progressively until an appropriately chosen cut-off criterion is met. Stationary probability vector can then be evaluated by an iterative method, such as that by Gauss-Seidel [43], which takes advantage of the sparsity and structure of . For each new value of , the previously computed vector is used as the initial solution to reduce the number of iterations required [42]. Thus, the numerical implementation of the approximation based on finite truncation implies the determination of an appropriate cut-off level . Here, we use the algorithm given by Artalejo et al. [36], the steps of which are described below.

For as the cut-off point, the modified generator will bewhere . Let be the stationary distribution of which satisfieswhere , by aggregating terms of the QBD , depending on levels. Define withAnd , . Here, is a row vector of dimension and is a row vector with dimension . By partitioning according to and , we havewhere is the matrix obtained by deleting the last column matrices and last row matrices from , , , and . These are block structured matrices with , , , and blocks, respectively. and are row vectors of dimensions and , respectively, with all entries equal to zero. From (12), we find thatFurther, we can have whereThe inverse of matrix can be determined, using methods given in Hunter [44] aswhere As the dimensions of the matrices increase in each iteration, the calculations to compute the above matrices involve multiplications and inversion of increasingly large matrices. If we exploit the structure of matrices in the above equations, we notice that the sparse blocks of and simplify the calculations. has only one nonzero square matrix of dimension in the last rows and has one in the last columns. So can be written in simplified form as . Further, becomes . These substitutions make the remaining operations in and simple, as they involve multiplications and inversion of only known simple matrices of size . The key step is to compute the matrix . The inverse in the definition of can be computed by using small-rank adjustment; that is, if we have the inverse of a matrix and we want the inverse of its adjustment , where is a matrix of smaller order than , then we have Here, we have , , , and . Thus, we obtain that so is obtained by multiplications and additions of already computed matrices. Finally, we haveSo, the computation of vector is reduced to solving system (14) subject to the normalization conditionwhere and are column vectors of dimensions and , respectively, with all entries equal to one. Finally, the vector can be solved substituting in (13). To get the cut-off value, successive increments of are made, starting from , and we stop at the point whenwhere is an infinitesimal quantity and is the infinity norm. The whole method of computing the stationary distribution using the finite truncation method is summarized in Algorithm 1.