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By Uggappakodi Narayan Bhat

ISBN-10: 3662388014

ISBN-13: 9783662388013

ISBN-10: 3662397064

ISBN-13: 9783662397060

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

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12 Let N(t) and n a) > be the number of arrivals in For, x > 0 , u > 0 0. Pr{W(t) = 0 , N(t) n -H A tn = e n! b) (O,t). 106) x} niW(O) n = K (t+u-x,t) - L oft d K (T+u,T)G (O;x,t-T). 103). 8 The queue M/G/1 with balking [Tak~cs (1961)]: For convenience consider the queue M/G/1 with unit arrivals. Customers arrive in a Poisson process and join the queue with probability otherwise. 1 if the server is free and with probability p(

90) for page 103]. 12 Let N(t) and n a) > be the number of arrivals in For, x > 0 , u > 0 0. Pr{W(t) = 0 , N(t) n -H A tn = e n! b) (O,t). 106) x} niW(O) n = K (t+u-x,t) - L oft d K (T+u,T)G (O;x,t-T). 103). 8 The queue M/G/1 with balking [Tak~cs (1961)]: For convenience consider the queue M/G/1 with unit arrivals. Customers arrive in a Poisson process and join the queue with probability otherwise. 1 if the server is free and with probability p(

N (n+j-i)! As the Poisson probabilities in the last expression and hence we have lim t-+oo 0 P .. 57) as follows. (t-T) = A ft p. (t-T) I b op ,(T)dT r 0 r 10 rJ rJ r=l r=l I T 00 = AOf + A ft P. (t-T) I r=l P. (t-T) 10 b op ,(T)dT rJ r 00 T I b op ,(T)dT r rJ r=l 10 (say). In these, first let T-+ oo• and then t (1. 70) then ' 00 I -+ A lim p. (t) of 1 t-roo 10 L 00 b r=l r 0P r j (T)dT 00 I r=l = b r 0P r if p j(T)dT (1. 71) 0 and 1 < ifp~l. 00 I2 ~ ft T ' 1\ (1. (t)dt rJ - 0f 00 -At oo I (O,oo), oo I n=O n+j-1 (k+l) (At)k 2.

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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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