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

Thaga Keaogile¹, Adebayo Fatai Adewole^1, and Sivasamy Ramasamy¹

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

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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References:

[1]	J. Medhi, Stochastic Models in Queueing Theory, 1: 01-457, 2003.
[2]	O.J Boxma, Q. Deng and A.P Zwart, “Waiting time asymptotic for the M/G/2 queue with heterogeneous servers’, Queueing Systems, 40: 5-31, 2002.
[3]	P. Hoksad, “On the steady state solution of the M/G/2 queue”, Advanced applied probability}, 11, 240-255. 1979.
[4]	D. Efrosinin, “Controlled Queueing Systems with Heterogeneous Servers: Dynamic Optimization and Monotonicity Properties of Optimal Control Policies in Multiserver Heterogeneous Queues”, 2008.
[5]	B.Krishnamoorthy, “On Poisson Queue with two Heterogeneous Servers”, Operations Research, 2 (3), 321-330, 1962.
[6]	V. P. Singh “Two-Server Markovian Queues with Balking: Heterogeneous vs. Homogeneous Servers”, Operations Research, 18 (1), 145-159, 1968.
[7]	J. H. Kim, H.-S. Ahn, and R. Righter, “Managing queues with heterogeneous servers”, J. Appl. Probab., 48 (2), 435-452, 2011.
[8]	Krishnamoorthy, B. and Sreenivasan, S. “An M/M/2 queue with Heterogeneous Servers including one with Working Vacations”, International Journal of Stochastic Analysis, Hindawi publishing company, 2012.
[9]	K. Senthamaraikannan, and R. Sivasamy, “Markov Renewal Queues of a M/M(a,d)/1/(b,N) System-Part I and Part II” Optimization, , Vol. 40, pp 121-145, 1997.
[10]	K. Senthamaraikannan and R. Sivasamy, “Embedded Processes of a Markov Renewal Bulk Service Queue”, Asia Pacific Journal of Operational research, 11 (1), 51-65, 1997.
[11]	H.Takagi, “Queueing Analysis-A Foundation of Performance Ealuation, Vol. 3. Discrete-Time systems( Elsevier Science Publishers, Amsterdam)
[12]	H.Bruneel and B. Kim, “Discretetime models for communication systems including ATM. Kluwer Academic Publishers, 1993.
[13]	R. Sivasamy and P M Kgosi, “An M/M+G/2 queue with heterogeneous machines operating under FCFS queue discipline, International Journal of Engineering and technical Research, (2014).