New PDF release: A Study of the Queueing Systems M/G/1 and GI/M/1

By Uggappakodi Narayan Bhat

ISBN-10: 3662388014

ISBN-13: 9783662388013

ISBN-10: 3662397064

ISBN-13: 9783662397060

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Extra info for A Study of the Queueing Systems M/G/1 and GI/M/1

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M= 0 (At) kl k b(k) B (t) n+j-i n It dGi(n-m)(T)e-A(t-T) 0 1j (A(t-T)]k b(k) B (t-T) k=O kl m+j m and *p 1].. (t) 1J Expressions for *P0 j(t) (1. 83) a> + A0It P. 0 r=l r rJ *Pi 0 (t) and follow (i ~ 1)(1. 84) by similar arguments. 7: For i ~ 0, j It T=O- + T=O- f It > 0, l d*Pir(T)e-A(t-T) L r=l s=T j~r L k=O [A(t-T)]k b(k) k'· j- r B(t-T±y)-B(t-T)] 1-B(t-T) It d*P. (T) Ae-A(S-T) dse-A(t-T) 10 [A{t-Tl] k! k=O k b (k+l) j+l B{t-s+vl-B{t-s)] 1-B(t-s) (1. 85) 33 Proof: Consider the last departure point and after this departure.

N+j-i n~l i (mb)m-i (n+J'-i)! - e L. m (m-i)! m=l. t ( t-mb) n-m+j (n-m+j)! llil 1 n. '( t - nb) A x. 113) Further, we get ~ 0). 57) we obtain x x-n = I I n=O k=O x >.. (k+j-1)! (k-m+j)! t(mb)m-1 kbfkb+b (T-mb)k-m+j. 115) bt ] . Finally, for the steady state probabilities, we get 1 - p P*0 00 P* = (1 - p) j where I n=O -np [e (np)n+j-1 -np {ne2n+j e (n+j)! ] (n+j-1)! b. (b) The queue M/Ek/1: Let the service time distribution be given by the k-Erlangian, k-1 - -kllt (kllt2 dB(t)- e (k-l)' klldt.

Inf {tiQ(t) = 0} , Q(O) = i, (2. 5) In this system we shall first obtain these probabilities for and then extend the results for i > 0. i=O, It may be noted, the queues M/G/1 and GI/M/1 may be called dual systems and the duality relations existing between structures of the queue length processes of the two systems will be fully exploited in our treatment. In view of this we have maintained the duality even in our definitions. 2 The Busy Period T 0 Consider a busy period that begins with the arrival of a customer.

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A Study of the Queueing Systems M/G/1 and GI/M/1 by Uggappakodi Narayan Bhat


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