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(c) Use partial-fraction expansion to show that p(i, s) = s (z2 ) i for i = 1, 2, and s = 0, 1 Specify the values of 0 and 1 421 Consider a multi-server queueing system with c unreliable servers Jobs arrive according to a Poisson process with rate The required service times of the jobs are independent random variables having a common exponential distribution with mean 1/ The service of a job may be interrupted by a server breakdown The server operates uninterruptedly during an exponentially distributed time with mean 1/ It takes an exponentially distributed time with mean 1/ to bring a broken-down server to the operative state Any interrupted service is resumed at the point it was interrupted It is assumed that an interrupted service is taken over by the rst available server Denote by p(i, s) the limiting probability of having i jobs present and s operative servers for i 0 and 0 s c Prove that the probabilities p(i, s) can be computed by using the geometric tail approach In particular, verify that p(i, s) s i as i
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for a constant s , where is the reciprocal of the smallest root of det [M(z)] = 0 on the interval (1, ) Here M(z) = (mst (z)), s, t = 0, 1, , c is a tridiagonal (c + 1) (c + 1) matrix with mss (z) = z [ + s( + ) + (c s) ] + s /z, ms,s 1 (z) = (c s + 1) and ms,s+1 (z) = (s + 1) This problem is based on Mitrani and Avi-Itzhak (1968) 422 Consider the unloader problem from Example 412 again Assume now that the unloading time of a ship has an Erlang (L, ) distribution and the repair time of the unloader has an Erlang (R, ) distribution Letting = ( L/ )(1 + R/ ), it is assumed that the server utilization is less than 1 Interpret the unloading time of a ship as a sequence of L independent unloading phases each having an exponential distribution with mean 1/ Also, interpret the repair time of the unloader as a sequence of R independent repair phases each having an exponential distribution with mean 1/ Let state (i, 0) correspond to the situation the unloader is available and i uncompleted unloading phases are present (i 0) Let state (i, r) correspond to the situation that there are i uncompleted unloading phases (i 1) and the unloader is in repair with r remaining repair phases (1 r R) Denote by p(i, s) the equilibrium probability of state (i, s) and de ne the generating functions Gs (z) by G0 (z) = p(i, 0)zi and Gr (z) = p(i, r)zi for |z| 1 i=0 i=1 (a) Verify that Gs (z) = det As (z) , det A(z) s = 0, 1, , R
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Here A(z) is the (R + 1) (R + 1) matrix A(z) = (1 z)M z(1 zL )I + zQT , where M = diag( , 0, , 0) and QT is the transpose of the transition matrix Q = (qij ) with q0R = q00 = , qi,i 1 = qii = for 1 i R and the other qij = 0 The matrix As (z) results from replacing the (s + 1)th column vector of A(z) by the vector b(z) with bT (z) = (( (1 z) z)p(0, 0), 0, , 0) (b) Conclude that for any s = 0, 1, , R, p(i, s) s i as i
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for a constant s , where is the reciprocal of the smallest root of det A(x) = 0 on the interval (1, ) Note that for Erlangian service the polynomial equation det A(z) = ( 1)R+1 [{ z(1 zL ) (1 z) + z}{ z(1 zL ) + z}R z( z)R ] = 0 is obtained by expanding det A(z) in the cofactors of its rst row
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