By Uggappakodi Narayan Bhat

ISBN-10: 3662388014

ISBN-13: 9783662388013

ISBN-10: 3662397064

ISBN-13: 9783662397060

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

**Sample text**

R-i-s r r=i+s k=O 00 0 Jt e -T(t-T) (;\(t--r)] (n-1)! (t-T)] (n-1)! +n r dB (T) r r-i-n (h)k r-~-k I - k! - L.. +l r dB (-r) r (i ( ) ( ) +s) b k. b n r-1-s s r-i (;\(t--r))n-1 rii (AT)k-n I k=n (k-n)! n=l (n-1)! b(k~ [i + (r-i)n] r-1 k 23 r-i 00 L ..!. (n-1)! which after some simplifications gives k-1 rt (k-1)! ~(k~ r-~ (r-i)(k-l)t k! 49), after some rearrangement. 4: For i ~ oo p. 51) Proof: Case (i): i=O The transition 0 + 0 in time (O,t] can occur, either with no arrival or with at least one arrival in case let 1(0 < 1 < t) for the last time.

96) [Fig. (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}.

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. *P. 1r (T) T in [O,t) and transitions before Before this departure transitions are of the type and after the departure the transitions are of the type is a departure point). 78) we have the theorem. 85). In this and the following theorem, we shall not be simplifying the expressions derived by probabilistic arguments. easier to simplify in special cases.

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

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