Solving Multiple-Server Queues in Rough Environment

M. Thangaraj,P. Rajendran

Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India.

*Corresponding Author E-mail: mthangaraj51@gmail.com, prajendran@vit.ac.in

ABSTRACT:

In this paper, a multiple-server queue in rough environment is considered in which the arrival rate and service rate are rough interval numbers. A new method namely, multi-level method is proposed to determine the performance measures in the given queueing system as rough interval numbers. The solution procedure for the proposed method is demonstrated with the help of numerical example. The performance measures of the multiple-server queue in rough environment provided by the proposed method are rough interval numbers which are very much useful for decision makers for taking the managerial decision-making.

KEYWORDS: Multiple-server queue, Rough interval numbers, Performance measures, Rough interval little formula, Multi-level method.

INTRODUCTION:

Queueing models [1, 2] are mathematical models which have wide applications in service organizations such as healthcare services as well as manufacturing firms, in that various types of customers are serviced by various types of servers according to specific queue disciplines. The inter-arrival time and service time of the queueing model are required to follow certain crisp probability distributions. In computer systems, the concepts of queueing theory are quite useful to estimate the value of some computer performance measures. However, in many practical applications the distribution of inter-arrival time and service time are not crisp values always. Thus, an imprecise queueing model is much more realistic and has wider applications than the usual crisp queueing models. Problems of multiple-server interval queues are often encountered in service industry, such as in hospital clinics.

Li and Lee [3] developed analytical based solution procedure for solving two fuzzy queueing models based on the fuzzy theory and the possibility concept. Two different approaches to analyze fuzzy queues were discussed by Negi and Lee [4].

Chiang Kao et al. [5] solved four simple fuzzy queues with a single server using fuzzy theory and the parametric programming technique. Shu Wang and Dingwei Wang [6] presented a solution procedure for solving multiple-server fuzzy queuing system using fuzzy theory and the parametric programming technique. Pandian and Rajendran [7] developed new approach to determine the performance measures of multiple-server fuzzy queues in which all parameters are fuzzy numbers.

Kembe et al. [8] studied the queuing characteristics at the Riverside Specialist Clinic of the Federal Medical Centre, Makurdi was analyzed using a multi-server queuing model and the waiting and service costs determined with a view to determining the optimal service level. Mohammed Shyfur Rahman Chowdhury et al. [9] described several common queuing situations and present mathematical models for analyzing waiting lines. Zhao et al.[10] studied a queue-based interval-fuzzy electric-power system model which is developed through coupling fuzzy queue theory with interval-parameter programming .

Hajnal Vassa and Zsuzsanna K. Szabo [11] developed an M/M/3 queue model to characterize the patient flow in the Emergency Department situated in Mures County, Romania as a case study. Vijay Prasad et al. [12] derived the total cost with assumption of certain waiting cost in both cases and the expected total cost is less for single queue multi server model as comparing with multi queue multi server model. Tiwari et al. [13] have described the M/M/s queue model to solve waiting line and to minimize estimated total cost. Fathi-Vajargah and Ghasemalipour [14] have obtained a theorem concerning the average chance of the event “r servers (r≤ k) are busy at time t”, provided that all the servers work independently and simulated the average chance using fuzzy simulation method and obtain some results on the number of servers that are busy.

2. Preliminaries

We need the following definitions of the basic arithmetic operators and partial ordering on a set of all rough intervals which can be found in Hongwei Lu et al.[15] .

Multiple-Server Queueing Model in Rough Environment:

Consider a multiple-server queueing in rough environment model with single-phase structure in which arrival rate and service rate are rough interval numbers. The following assumptions are coupled with the considering model:

(i) The number of service channels is s and the service channels are identical.

(ii) The arrival source population and the queue are infinite.

CONCLUSION:

A multiple-server queue model with infinite capacity in rough environment is considered in this paper. The performance measures of multiple-server rough interval queue model are obtained as interval numbers using the multi-level method. The proposed method provides ranges of the performance measures of the considered system. Numerical example is presented to illustrate the solution procedure of the multi-level method. The rough interval performance measures of the rough interval queue with infinite capacity obtained provide more insight for managerial decision-making.

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