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where an is the same as the constant an in Theorem 921 except that B(t) is replaced by R R Bi (t), i = 1, 2 Also, argue that 1 p0 = { 1 j =0 p1j + 2 (1 j =0 p1j )}, where i is the mean of the distribution function Bi (Hint : note that the long-run fraction of service completions at which j customers are left behind equals the long-run fraction of customers nding j other customers present upon arrival) 94 Consider the M/G/1 retrial queue from Exercise 233 again Let p0j (p1j ) denote the long-run fraction of time that the server is idle (busy) and j customers are in orbit for j = 0, 1, (a) Use the regenerative aproach to establish the recursions j p0j = p1,j 1 , p1j = aj 1 p00 + 1 a0 1 a0
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where ak = 0 e t ( t)k (1/k!){1 B(t)} dt with B(t) denoting the probability distribution function of the service time of a customer (Hint : let T0j (T1j ) denote the amount of time during one cycle that the server is idle (busy) and j customers are in orbit and let N0j denote the number of service completions in one cycle at which j customers are left behind in orbit Argue that E(T1,j 1 ) = E(N0j ) for j 0, E(T1,j 1 ) = j E(T0j ) for j 1
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of time that j customers are in orbit during a given service time when k customers were left behind in orbit at the completion of the previous service time) (b) Use generating functions to verify that p00 = (1 ) exp 2 1 (z) dz , 0 1 (z)
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where (z) = 0 e t (1 z) {1 B(t)} dt (c) Instead of the M/G/1 queue with a linear retrial rate, consider the M/G/1 queue with a constant retrial rate That is, retrials occur according to a Poisson process with rate when the orbit is not empty Modify the above results This problem is based on De Kok (1984)
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95 Consider the M/G/1 queue with exponential rst service from Exercise 91 again Assume that service is in order of arrival Let Wq (x) denote the limiting distribution function of the delay in queue of a customer (a) Verify that the generating function P (z) = j =0 pj zj is given by P (z) = p0 [1 ( (z) z 0 (z))] , 1 (z)
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( ) where E(zLq ) = p0 + 1 [P (z) p0 ] Next prove that z
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where (z) = 0 e (1 z)t {1 B(t)} dt and 0 (z) = 0 e (1 z)t {1 B0 (t)} dt (b) Verify that the relation (2514) also applies to the M/G/1 queue with server vacations,
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e sx {1 Wq (x)} dx =
p0 (1 b0 (s)) 1 1 p0 , s s + b (s)
where b0 (s) = 0 e sx b0 (x) dx is the Laplace transform of the density of the exceptional rst service and b (s) = 0 e sx b(x) dx is the Laplace transform of the density of the ordinary service
96 Consider again the M/G/1 queue with server vacations from Exercise 92 Assuming that service is in order of arrival, let Wq (x) denote the limiting distribution function of the delay in queue of a customer (a) Letting P0 (z) = j =0 p0j zj and P1 (z) = j =1 p1j zj , verify from the recursion scheme in Exercise 92 that P0 (z) = 1 (z) (z) and P1 (z) = zP0 (z) , E(V ) 1 (z)
where (z) = 0 e (1 z)t {1 V (t)} dt and (z) = 0 e (1 z)t {1 B(t)} dt Argue ( ) that relation (2514) also applies to the M/G/1 queue with server vacations where E(zLq ) is given by P0 (z) + P1 (z)/z (b) Verify that the Laplace transform of 1 Wq (x) is given by 0