. Cycle-time distribution in cyclic exponential network.

Consider an M-node cyclic network with K circulating jobs, the nodes having independent exponential servers with parameters ( ,, i = 1, 2 ,..., M. Let W; be the sojourn time at node i and let T = W + . . . + WM be the cyclic time (sum of consecutive sojourn times). Show that the LST of T is given by E [exp(-sT)] = E{exp[-s(W, + .. . + WM)]}

M

=

M. +1 1

= > a(m ..., M) IT Re s ≥ 0,

\1 +s/ui 1, EC where M C = C(M, K- 1) = ("1, ..., "M) : " ≥ 0, > ni = K -1} and i=1 M a(n) ,..., "M) =]]
Find E (T) for M = 2 (Schassberger and Daduna, 1983). Note that the distribution has a product form.