American Journal of Applied Mathematics and Statistics
ISSN (Print): 2328-7306 ISSN (Online): 2328-7292 Website: http://www.sciepub.com/journal/ajams Editor-in-chief: Mohamed Seddeek
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American Journal of Applied Mathematics and Statistics. 2015, 3(2), 54-58
DOI: 10.12691/ajams-3-2-2
Open AccessArticle

Geo (λ)/ Geo (μ) +G/2 Queues with Heterogeneous Servers Operating under FCFS Queue Discipline

Thaga Keaogile1, Adebayo Fatai Adewole1, and Sivasamy Ramasamy1

1Department of statistics, University of Botswana, Gaborone, Botswana

Pub. Date: March 06, 2015

Cite this paper:
Thaga Keaogile, Adebayo Fatai Adewole and Sivasamy Ramasamy. Geo (λ)/ Geo (μ) +G/2 Queues with Heterogeneous Servers Operating under FCFS Queue Discipline. American Journal of Applied Mathematics and Statistics. 2015; 3(2):54-58. doi: 10.12691/ajams-3-2-2

Abstract

This article discusses the steady analysis of a discrete time queue of Geo/Geo+G/2 type. All arriving customers are served either by server-1 according to a geometrically distributed service time S1=k slots for k=1,2, …∞, with mass function f1(k)==Pr(S1=k) = μ(1- μ) k-1 with mean rate 0<μ<1 or by server-2 with a general service time S2= k for k=1,2, …, with mass function f2(k)==Pr(S2=k) with mean service time is or mean service rate μ2=1/β. Sequel to some objections raised on the use of the classical 'First Come First Served (FCFS)' queue discipline when the two heterogeneous servers operate as parallel service providers, an alternative queue discipline in a serial configuration of servers are considered in this work; the objective is that if, in a single-channel queue in equilibrium, the service rate suddenly increases and exceeds the present service capacity, install a new channel to work serially with the first channel as suggested by Krishnamoorthy (1968). Using the embedded method subject to different service time distributions we present an exact analysis for finding the ‘Probability generating Function (PGF)’ of steady state number of customers in the system and most importantly, the actual waiting time expectation of customers in the system. This work shows that one can obtain all stationery probabilities and other vital measures for this queue under certain additional and simple but realistic assumptions.

Keywords:
poisson arrival service time distribution pgf of queue length distribution waitime distribution mean queue length and mean waiting time

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