Comparison of queues with different discrete-time arrival processes (Q2712999)

Consider two stochastic event graphs with the same potential arrival epochs \(\{T_n\}\) (coupled say by a uniformization procedure), but where the actual numbers of arrivals at time \(T_n\) differ, say \(A^1_n\), \(A^2_n\) are admitted where the sequences \(\{A^1_n\}\), \(\{A^2_n\}\) have the same average behaviour. Let \({_aW^i_k}\) be the traveling time of the \(k\)th arrival in the \(i\)th graph to some fixed transition \(q\) (for example, the total sojourn time of the \(k\)th customer in a tandem queue with \(q\) stations) and \({_bW^i_k}\) the traveling time of the \(k\)th potential arrival at \(T_k\). It is shown that if \((A^1_1,\dots, A^1_n)\) is smaller than \((A^2_1,\dots, A^2_n)\) in the convex ordering, then \(({_aW^1_1},\dots, {_aW^1_n})\) is smaller than \(({_aW^2_1},\dots, {_aW^2_n})\) in the increasing convex ordering, and similar results are given for the \({_bW^i_k}\). This relates to and generalizes a set of questions going back to Ross. As applications, it is shown that the system with two identical independent sources performs better that the one with identical arrival epochs, that fixed batch sizes are better than random batch sizes, and that fluid scaling improves the performance. The analysis is carried out in the dense class of Markovian arrival processes.