NOPASSING MULTISERVER QUEUE WITH ADDITIONAL HETEROGENEOUS SERVERS AND INTER-DEPENDENT RATES Madhu Jain Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee-247667, India E-mail:
[email protected]
G.C. Sharma School of Mathematical Sciences, Khandari, Agra-282004, India E-mail:
[email protected]
Abstract In many congestion problems, the customers may not like to join a long queue and may balk. To reduce the balking behavior of the customers, the provision of additional servers can be made in such situations. In the present paper, we investigate a Markovian queueing model with balking, interdependent rates and additional servers under no passing constraint. Due to nopassing restriction, the customers are allowed to depart from the system in the same chronological order in which they entered the system. Based on the service requirements, the customers are classified in two classes; the type ’A’ of the customers has zero service time whereas the service times of type ’B’ customers follow exponential distribution. By using recursive approach, we obtain the queue size distribution at equilibrium. Various performance indices such as the mean waiting time and the difference between the expected waiting time for both types of customers have been established. By setting appropriate parameters, some special cases of existing models have been discussed. In order to demonstrate the computational tractability of the performance indices derived analytically, numerical experiment has been performed.
Keywords : Markovian queue, Interdependent rates, Balking, No passing, Additional heterogeneous servers, Queue size distribution, Expected waiting time. 1. Introduction Performance modeling of Markov queue with balking has attracted many queue theorists due to its applications in real life congestion problems. However, sometimes due to physical restriction or otherwise, the customer has to
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depart from the system in the same chronological order in which he entered; this may be the case even if he does not require the service. To illustrate, we cite examples of security check at airport, parking lot with single exit point, message transmission through secure channel, etc. The waiting time of the customers even they don’t require service in case of no passing multiserver queueing model was first time studied by Washburn [1]. The other recent past worth-mentioning works in this direction are due to Sharma et al. [2] and Jain et al. [3]. The time- sharing M/M/φ(.) queueing model with no passing was developed by Jain and Prem Lata [4]. Jain et al. [5] examined a multiserver loss-delay queueing system with priority and nopassing. No passing concept for finite population loss - delay queueing system and M/M/m/K queue with additional server was considered by Jain [6] and Jain and Ghimire [7]. Later on, Jain et al. [8] provided performance indices of Markovian loss and delay queueing model with no passing and removable additional servers. Performance prediction for Markovian loss and delay queueing model with no passing and removable additional server was analyzed by Jain and Singh [9]. Ma et al. [10] considered a discrete time queueing system with two classes of customers and priorities. The first class of customers are with delay and loss, and the second class of customers are with loss. The second class of customers have preemptive resume priority to the first class of customers. Using the method of matrix analysis, the mean queue length, the loss probability and the system utility for the two classes of customers are derived. Fan [11] developed a queueing model for mixed loss-delay systems with general inter arrival processes for wide-band calls. Vuyst et al. [12] analysed a discrete-time queueing model with packet arrivals that are either delay-sensitive (type 1) or delay-tolerant (type 2). The prominent feature of this model is its reservationbased queueing discipline, which reduces the queueing delay perceived by the packets of type 1 at the cost of allowing higher delays for the packets of type 2. Jain and Jain [13] proposed a modified diffusion approximation for loss and delay GX /GY /R queueing system with discouragement. Orallo and Carb [14] examined network queue and suggested loss analysis using histogram-based traffic models.
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A few papers has appeared on queueing problems as for as interdependent processes and correlated rates are concerned. The busy period of a correlated queue with exponential demand and service was studied by Borst and Combe [15]. Gray et al. [16] developed an M/G/1 type queueing model with service times depending on queue length. The M/M/1 interdependent queueing model with controllable arrival rates was proposed by Srinivasa Rao et al. [17]. They observed that the mean dependence rate between the arrival and service processes can be helpful in reducing the congestion in queues and delays in transmission. Begum and Maheswari [18] studied M/M/c interdependent queueing model with controllable arrival rates. Jain and Sharma [19] considered the controllable queue with balking and reneging. To reduce the balking behavior of the customers in the controllable queue, the provision of additional removable servers was made by Jain and Sharma [20]. Sitrarasu et al. [21] developed the M/M a,b /C interdependent queueing model with controllable arrival rates. A single server interdependent queueing model with controllable arrival rates and reneging was examined by Singh et al. [22]. Jain et al. [23] have examined the controllable and interdependent rates for the machine repair problem (MRP) with additional repairman and mixed standbys. Yang et al. [24] described the optimization and sensitivity analysis for controlling the arrivals in the queueing system with single working vacation. Jain et al. [25] have presented the queueing analysis of maintenance float system with mixed standbys and interdependent rates. In this investigation, Markovian queue with inter-dependent rates and balking under the restriction of no passing restriction in analyzed. By introducing the queue dependent heterogeneous additional servers who turn one by one according to threshold policy, the balking behavior of the customers is controlled. The rest of the paper is structured as follows. In section 2, we describe the model by stating the assumptions and notations. The steady state queue size distribution using recursive method is obtained in section 3. Section 4 is devoted to the analysis of finite capacity and finite population models. The expression for the expected waiting time and difference between expected waiting for both types of customers are derived in section 5. By taking an illustration, numerical results are facilitated in section 6. The final section 7 contains the concluding remarks and directions for further research work.
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2. Model Description
The multi-server Markovian queueing model to explore the no passing and balking concepts is considered. To formulate the mathematical model the following assumptions are made:
(1) The customers arrive in the system in Poisson fashion with rate λ . In case when all permanent as well as j th (j = 0,1,,r) additional servers are busy, the customers may balk with probability βj = 1 − βj . (2) The service of the customers are rendered according to exponential distribution by a service facility consisting of R homogeneous permanent servers along with r additional removable heterogeneous servers according to following rule: (i) If there are n < R customers in the system then only n permanent servers provide the service with rate µ ; the permanent server switches over to faster rate µf in case when all permanent servers are busy. (ii) If there are n (R ≤ n < N1 ) customers in the system, then all permanent servers provide service to the customers with faster rate . (iii) If there are greater than or equal to Nj and less than Nj+1 (i.e. Nj ≤ n < Nj+1 ) customers in the system then all permanent and j additional servers will provide service. Again the j th (j=1, 2,,r-1) server will be removed as soon as the queue size ceases to Nj . (iv) If Nr customers accumulate in the system then all permanent and additional servers will provide service; and rth additional server will be removed as soon as queue size ceases below Nr . (3) In order to determine the waiting time for the no passing model, we assume that the system has two types of customers i.e. type A and type B customers. The proportion of type A and B customers are p and 1-p, respectively. The service time of type A customer is zero whereas type B customers require service which is exponentially distributed. Let NA and NB be the random variables denoting the number of customers of type A and B, respectively.
The state dependent arrival rate for finite capacity model is given by
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1≤n
