Effective upper bounds for expected cycle times in tandem queues with communication blocking (Q2748551)

A tandem queueing system with \(K\) stations and communication blocking is considered. It is assumed that processing times at each station are mutually independent and identically distributed. In order to derive upper bounds for the cycle time of the original system the synchronous system is introduced. Denoting the set of stations with possible processing time by \(\Omega =\{i_1,\ldots,i_m\}\) \((1=i_1<\ldots<i_m=K)\) a set of feasible pairs of groups is introduced by the relation: NEWLINE\[NEWLINE\begin{multlined} \Phi= \left\{(A,B): A \cap B=\emptyset,\;A \cup B = \Omega, \right. \\ \left. \text{ and if } i_{k+1}=i_k+1, \text{ then either } i_k\in B, i_{k+1}\in A, \text{ or } i_k\in A, i_{k+1}\in B\right\}.\end{multlined} NEWLINE\]NEWLINE Using this notation the following upper bound for the expected cycle time is found, NEWLINE\[NEWLINE E[C] \leq \min_{(A,B)\in\Phi} C_{U}(A,B), NEWLINE\]NEWLINE where NEWLINE\[NEWLINE C_{U}(A,B) = E\Bigl[\max_{i_k\in A} I_k\Bigr] + E\Bigl[\max_{i_k\in B} I_k\Bigr] NEWLINE\]NEWLINE is the expected cycle time in the synchronous system with group \((A,B)\), \(I_k\) is a random variable representing the processing time at \(k\)th station. To make this bound effective, an order on the family of pairs is introduced. By using this order an effective upper bound for the expected cycle times in tandem queues with communication blocking is derived.