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

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.


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 PHdistributed 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.

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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 (0, Q, N, M) 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 continuoustime Markov chain whose generator matrix is a leveldependent 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.

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 e  = 1 and  0 = −e  , where e  is the column vector of dimension  with all the entries equal to one.The matrix  is a nonsingular square matrix with ()  < 0, 1 ≤  ≤ , and ()  ≥ 0, 1 ≤  ̸ =  ≤ .The matrix  0 is a nonnegative, -dimensional column vector, grouping the absorption rates from any state to the absorbing one.The matrix  0  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 Exp(  ).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 Exp() distribution.During a WV period of a server, if a customer arrives to that server, it will serve the customer with Exp( V ) distribution, where  V <   ; 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  V 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  V ; that is, if the waiting customer finds all the servers in their WV period, the customer activates an impatient timer .This impatient timer  follows Exp() 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.

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For  ≥ , The matrix  is an identity matrix of dimension .Here, for 0 ≤  ≤  − 1, dimension of the matrices  () 0 ,  ()  1 , and  () 2 increases with the levels; and for  ≥ , the matrices are of dimensions ( + 1) × ( + 1) 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.
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 leveldependent 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  = 0 and e = 1 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.

Performance Measures
The performance measures give the qualitative behavior of the model under study.In a multiserver queueing model, the efficiency of the model depends upon the mean number of busy servers, the mean queue length, the blocking probability, and the mean number of customers lost due to impatience.
In our model, the server serves even during its vacation.Therefore, the number of busy servers will be , 0 ≤  ≤ , if there are  customers in the system and when the system has more than  customers, all the servers will be busy serving customers either in WV or in nonvacation with rates  V and   , respectively.The mean number of servers busy in nonvacation is The mean queue length of the system under study is Availability of the server, , is the probability that an arrival finds a server free.It can happen only if the number of total customers in the system in less than  and is given by The blocking probability of a multiserver queue is the probability of refraining a customer from service.In our model, a customer is kept waiting in the queue for service when all the servers are in busy state, either in WV or in nonvacation, that is, when the number of customers in the system is more than : The mean number of customers lost by the system is the average of customers who have abandoned the system as a result of waiting in a queue ( > ) with all servers in WV ( = 0); therefore,

Numerical Examples
Let us illustrate the behavior of our PH/M/c/WV queue with the help of some numerical examples.Algorithm 1 is coded in MATLAB © .The algorithm computes the stationary distribution and its main objective is to find the termination criteria of the level   .We start with an initial value  ≥  + 1 and progressively increase the value of  until a change in the stationary probability  is sufficiently small due to increased .We choose the smallest value of   such that max 0≤≤  ‖(  , ) − (  − 1, )‖ ∞ < , for  = 10 −6 .With this selection criterion, we find the values of   , the mean queue lengths, and the blocking probabilities for various sets of parameter values and for different arrival processes.Here, we take some examples (from Chakravarthy et al. [37]) of well known distributions and give their PH representations below: (1) Exponential (Exp): (2) Erlang-2 (Erl): (3) Hyperexponential-2 (Hyp): All these PH distributions have the same mean arrival rate  = 1.The standard deviations of the three distributions are 1.0, 0.70711, and 2.24472, respectively.The service rate during nonvacation period,   , is calculated for specific values of  using the formula  = /  .We have chosen  = 0.1, 0.5, and 0.9 for given values of  ( = 1,3,6).The effect of parameters on system performance is illustrated here.We will mention the models having interarrivals as exponential, Erlang, and hyperexponentially distributed as exponential model, Erlang model, and hyperexponential model, respectively.

Effect on Cut-Off
Value   .We have illustrated here the effect of the parameters, namely, traffic intensity (), rate of vacation duration (), service rate during WV ( V ), and the type of arrival process, on the truncation cut-off value   .
For three different values of  ( = 1, 3, 6), different tables are presented.The impatient rate is fixed at  = 0.1 for the tables.We have the following observations from Tables 1, 2, and 3: (1) The cut-off value increases with the increase in the variance of the distribution of the interarrival times.For Erlang model, the termination is the fastest whereas for the hyperexponential one it is the slowest.This behavior seems to be the same for all sets of parameter values and for all .(2) For a particular arrival process when the traffic load is small, the value of   decreases with increase in  V and also with the increase in .But for high  (> 0.5) and high , it shows the reverse property for all arrival processes and for any number of servers ; that is, when the system load is heavy and the system has small vacation duration, the cut-off value seems to be high for all types of arrival processes and any number of servers.
(3) When the vacation duration rate  is too high (=100),   value remains unaffected by vacation-service rate  V for any number of servers.
These observations show that the cut-off value depends on the system parameters and also on the arrival process but becomes independent of vacation-service rates when we have systems with small vacation duration.

Effect on Mean Queue Length
(1) The mean queue length of the system depends upon the arrival process.Tables 1, 2, and 3 show that systems with interarrival distributions of high variance have higher number of customers in the queue for any number of servers.We have fixed the impatient rate at  = 0.1.For  = 1, 3, with  = 0.5, we have Figures 1, 2, and 3, respectively, and with  = 0.9, we have Figures 4, 5, and 6, where the changes in mean queue lengths are given for increasing vacation-service rates.
A hyperexponential model always has the highest mean queue length compared to the corresponding Erlang and exponential models, irrespective of the number of servers.
(2) When the traffic load is heavy,  = 0.9, increase in vacation-service rate does not affect the mean (3) For  = 0.1 and  = 1, 3, 6, we plot Figures 7, 8, and 9, respectively.Here, the hyperexponential model has the least queue length compared to the corresponding Erlang and exponential models.For  = 6, the queue lengths are the same and all of them are shown in Figure 9. Also, it can be seen that, for increased vacation-service rates, arrival processes do not have much influence on mean queue lengths.impatient rate is small, the mean queue length for Erlang model becomes minimum.As the impatient rate increases, it shows the reverse behavior.The point of inflection depends upon the service rate  V .But the impatient rate does not have much effect on queue lengths when the arrival process is Erlang, whereas for hyperexponential model, the mean queue length decreases significantly with the increase in impatient rate.
Therefore, systems with hyperexponential arrivals have the longest queues compared to corresponding Erlang or exponential arrivals.For light loaded systems ( = 0.1) and highly impatient customers (1/ < 10), hyperexponential arrivals give the minimum queue lengths.When the customer impatient rates are small, the system behavior depends on the vacation-service rates.

Effect on Blocking Probability.
From the tables and the graphs plotted for blocking probability, we have seen the following properties for the models under study: (1) Figure 13 gives that for a single-server system the blocking probability of a hyperexponential model is minimum and that for Erlang is maximum while  = 1,  = 1, and  = 0.9.This behavior is also observed in multiserver models when  is too small.For higher values of , multiserver models follow the reverse nature; that is, Erlang model gives the minimum queue length and the hyperexponential one gives the maximum of those three different arrival models.That is, the chance of blocking a customer with Poisson arrival in a single-server as well as in a multiserver queue is always sandwiched between those with Erlang and hyperexponential arrivals.
(2) When we have a single-server Erlang model, the blocking probabilities seem to reduce up to 6% with an increased rate of service during vacation.Because of a single-server queue, the server rarely goes to vacation (as the systems are rarely empty) and the customers are served at a higher rate most of the times.And even when the system goes to vacation, because of the single-server queue, the queue started to form rapidly making the customers impatient and leave the system more often than a multiserver model.These contribute significantly to dropping the blocking probability in a single-server queue as seen in the plot.
(3) Figures 14 and 15 show that the hyperexponential model is not much affected by the vacation-service rate, whereas the Erlang model can reduce the blocking probability by up to 4% for increased vacationservice rates.A model with  = 1 and Erlang-2 arrival process has the maximum chance to make a customer wait in queue compared to exponential and hyperexponential arrivals, but as the number of servers is increased, hyperexponential arrival model has the highest blocking probability.The exponential arrival model always remains in between these two.

Average Number of Servers Busy in Nonvacation.
The mean number of servers that are in working status during nonvacation period is shown in subsequent figures: (1) Figure 16 is a plot of blocking probabilities with changing .Here, for an increase in vacation duration, the number of servers that remain busy is high, because the servers serve at a low rate but for longer time, and a new arrival will be served by an idle server if any.Consequently, it increases the number of busy servers in the system.
(2) Figure 17 shows that if the service rate is fast, customers are served at a faster rate which results in a lower number of busy servers in nonvacation period.This is true for all the three types of arrival models.
(3) The impatience makes a customer leave the system unserved and for high impatient rates more servers remain idle (Figure 18).But if the impatient rate is increased beyond a certain value ( > 6), the mean number of busy servers remains unaffected.

Average Customer Loss.
We plot the mean number of customers who abandon the queue without getting served in Figures 19 and 20.The values of  for these plots are  = 0.1 and  = 1, respectively, keeping the other parameters fixed for both cases.
When  = 0.1, that is, the system has longer vacations, the number of lost customers is less compared to the corresponding model for  = 1.In both cases, the hyperexponential models have the maximum customer loss, which is up to 60% more than the Erlang model.Also, the effect of impatient rates on customer loss is negligible for a system having small vacation duration.
For Figure 19, when vacation duration rate  = 0.  but minima for hyperexponential one at the point where impatient rate is equal to one.From Figure 18, we can see that when  = 3, the number of busy servers in nonvacation drops sharply until impatient rate becomes one and then it is almost consistent thereafter.This drop is more significant for hyperexponential model.This suggests that as impatient rates increase from zero to one, the number of busy servers in nonvacation period becomes less, which increases the probability of losing more customers for Erlang and exponential models.As the impatient rate increases beyond one, the number of busy servers in nonvacation remains consistent and the loss of customer is influenced mainly by the increased rate of impatience.But for the hyperexponential model, this behaviour alters because of the change in mean queue lengths with the change of impatient rates (Section 5.2(4)).Its influence can be seen more towards the point of inflection where the blocking probability of the hyperexponential model becomes the same as that of the exponential one.But as the impatient rate increases, the customers leave the system, increasing the customer loss.Thus, we have seen the role of various parameters on system performances and we are now in a position to handle them to enhance the system efficiency.

Conclusion
In this paper, we have analyzed the nonhomogeneous QBD model of a PH/M/c queue with impatient customers and  multiple working vacations.We have used the finite truncation method to determine the stationary distribution.The effects of system parameters on the performance measures of the model are illustrated with the help of some numerical examples.Comparisons are made for different interarrival time distributions and the effects of the parameters on those distributions are also presented.

Disclosure
The present address of Cosmika Goswami is School of Engineering, University of Glasgow, Rankine Building, Glasgow G12 8LT, UK.

( 4 )
The impatient rate  affects the queue lengths significantly, especially when  = 0.1.In Figures10, 11, and 12, the change in queue lengths with the increase in impatient rate is shown.When the
1, we can see local maxima for Erlang and exponential models