Diffusion Approximation Method for Multi-Server Queueing System With Balking

Abstract

This paper deals with the diffusion approximatien technique for solving piulti-server queueing problems with balking having Erlangian inter-arrival time and Erlangian service time distributions. Probability efjoining of a ' new customer to the system is assumed to vary as e'rv where T is a positive parametef and pt is the queue length. The ap'proximation technique is based on the theery of diffusion, considering only means and variances ofarriyai and 'departure processes, Approximate formultis for P (n), probability of finding n customers in the system, and L, mean number of customers in the system, at steady state, are given. Finally, compaiisons of approximate and exact or si:nulated values of mean number L of customers in the system are made for some EllEk!s (ee) systems with balking to show the effectiveness of the approximation technique and graphs of approximate values of L for several systems are drawn which can be used in practice.

Queueing Modelling of Machining System with Balking, Reneging, Additional Repairmen and Two Modes of Failure

“…Blackburn [3] gave optimal control of a single server queue with balking and reneging. Biswas and Sunaga [2] developed the diffusion approximation method for multi-server queueing system with balking. Abou-El-Ata and Shawky [1] derived the analytical solution of the single server Markovian over-flow queue with balking, reneging and an additional server for longer queues.…”

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“…The model is an extension of the M/M/s-consistent diffusion model for the GI/G/s queue developed by Kimura [13] where the base was a BD process. The model is applicable to queues with finite waiting spaces (Sunaga et al [19]; Yao and Buzacott [23]), finite sources (Biswas and Sunaga [3]; Halachmi and Franta [9]; Sivazlian and Wang [18] and so on; see, e.g., Biswas and Sunaga [2], Kimura [12] and Varshney et al [20] for other possible applications. Our model is also a refinement of the M/M/s-consistent model in the sense that it satisfies a conservation law for the steadystate distribution.…”

“…a ≡ λc 2 a + µc2 s , a * * k = a * * ≡ λc 2 a + µ and θ = a * * /a = (ρc 2 a + 1)/(ρc 2 a + c 2 s ). Substituting (4.7) into (4.6), we obtainp k = 1 − ρ, k = 0, ρ 1 − ρ ρk−1 , k 1,(4.8)with ρ = ζ θ .…”