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

Example text

When Pr{Q(t ) = 0 , t < t , T. > t 1 }. n n l. 27) the first busy cycle will then have the distribution R(n)(t) 0 52 We shall consider the two cases When n=l, all the i (~ 0) n=l and n > 1 waiting customers at t 0 separately. = 0 and the one arriving at this epoch should get served in one inter-arrival period [t 0 , t 1 ), say, in k batches. The probability of this event can be given as Ii dR~l)(t) 1 v=O i+l = i I I [b (k-1) \) k=l v=k-1 b (k)] V 00 b (k-1) I R-=i+l-v ft -AT (AT)k-l O e (k-l)! AdTdB(t).

1)]. Otherwise, let the last jump occur at probability of this event is AdT. 97) (n ~ 1) [Fig. (2)]. 95) is obtained by induction. it to be true for n = 0,1,2, ... 97) we get = e -At k k k k-1 k-1 A ft-x {t -(t-x) + t -(t-x) [y-(t-x)]} (k-1)! y=O k k-1 dxBk_ 1 (t-x-y)dB(y) = e -At Ak {tk-(t-x)k + tk-l_(t-x)k-l t-x (k-1) ! k k-1 [ k - (t-x) ]} (1. 95). 6). 4: For u > 0 , x > 0 and n Y*(O) = u > 0 as follows. n > 1 (O;x,t-T). ::. x , N(t) = n} = K (t+u-x,t). 101) T < t) at which u + T < X(T) < u + T + dT, n I m=l for the We therefore ft Pr{u + T < X(T) < u + T + dT, N(T) = m}.

J i (0 < t < oo), (t) and customers. Let so that = Bn(t)- Bn+l(t). 16) (n ~ 0) be a compound Poisson process with parameter Pr{D(t) = n} P.. 5) respectively and obtain the distribution of the busy period initiated by u (r=l,2 ... n) r o *(n) A. 18) T. = inf{tli + N(t) - D(t) ~ 0} . l. at which is non-Markovian; however, the points i + N(t) - D(t) The process T) (for some i + N(T) - D(T) = 0 are its points of For this process we give below two lemmas which are regeneration. 1. 1. 1: = (At)k b(k) e -At n+~-j dB (t) - L n n+i-j kl k=O n-j m+i - I I m=l k=l 0 It e -h (AT) kl k b(k) dB ( ) n T m+i dt Pr{N(t n -t m) - D(t n-tm) = j t -t n < m- t-T ' T > t -t }.

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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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