# Applied to derive the analytical solution of

Applied Mathematical Sciences, Vol. 2, 2008, no. 24, 1161 – 1167 On a Truncated Erlangian Queueing System with State – Dependent Service Rate, Balking and Reneging M.

S. El – Paoumy Department of Statistics, Faculty of Commerce, Dkhlia, Egypt Al-Azhar University, Girl’s Branch [email protected] Abstract The aim of this paper is to derive the analytical solution of the truncated Erlangian service queue with state-dependent rate, balking and reneging (M/ER/I/N (??, )) . We obtains nP,, the probabilities that there are “n” units in the system and the unit in the service occupces stage “s” (s = 1, 2,…, r ) .We treat this queue for general values of r , k and N. Keywords: truncated Erlangian service queue, balking, reneging.

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1 INTRODUCTION This paper considers the queueing system M/Er/1/N with state – dependent service rate, balking and reneging concepts .The Erlang distrbution, denoted by Er is a special case of the gamma distribution, is named after A.K. Erlang who pioneered queueing system theory for its application to congestion in telphone networks .The nontruncated queue: M/Er/1 was solved by Morse 3 at r =2 and white et al. 4 Who obtained the solution in the form of a generating function and the probabilities could be obtained by a power series expansion.

Al Seedy 1 gave an analytical solution of the queue:1162 M. S. El – Paoumy M/Er/1/N with balking only. This work had been followed by Kotb 2 who studied the analytical solution of the state – dependent Erlangian queue: M/Er/1/N with balking by using a very useful lemma.

In this paper we treat the analytical solution of the queue: M/Er/1/N ()??, for finite capacity considering by using a recurrence relations .We obtain s nP,, the probabilities that there are “n” units in the system and the unit in serive occupies stage “s” ( r s? ? 1 in terms of 0P . The probability of an empty system 0P is also obtained .The discipline considered is first in first out (FIFO). 2 THE PROBLEM ANALYSIS Consider the single – channel service time Erlangian queue having r – service stages each with raten?, with the state – dependent and reneging in the form: ?? ?? ?? ? + ? + ? =?? ??o t e t ft This means that the units are served with two different rates 1?r or 2?r depending on the number of units in the system whether k n??1 or N n k??+1 respectively.

Also, consider an exponential interarrival pattern with rate .n?Assume ()??1 be the probability that a unit balks (does not enter the queue). where :=?p(a unit joins the queue), ; 1 , 1 0N n??

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