Sojourn time in a single server queue with threshold service rate control

We study the sojourn time in a queueing system with a single exponential server, serving a Poisson stream of customers in order of arrival. Service is provided at low or high rate, which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, and otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace transform of the stationary sojourn time and describe a procedure to compute all moments as well. First we analyze the special case of continuous inspection, where the service rate immediately changes once the threshold is crossed. Then we extend the analysis to random inspection times. This extension requires the development of a new methodological tool, that is "matrix generating functions". The power of this tool is that it can also be used to analyze generalizations to phase-type services and inspection times.Comment: 16 pages, 13 figure

Reversibility and further properties of FCFS infinite bipartite matching

The model of FCFS infinite bipartite matching was introduced in caldentey-kaplan-weiss 2009. In this model there is a sequence of items that are chosen i.i.d. from $\mathcal{C}=\{c_1,\ldots,c_I\}$ and an independent sequence of items that are chosen i.i.d. from $\mathcal{S}=\{s_1,\ldots,s_J\}$, and a bipartite compatibility graph $G$ between $\mathcal{C}$ and $\mathcal{S}$. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In adan-weiss 2011 a Markov chain associated with the matching was analyzed, a condition for stability was verified, a product form stationary distribution was derived and the rates $r_{c_i,s_j}$ of matches between compatible types $c_i$ and $s_j$ were calculated. In the current paper, we present several new results that unveil the fundamental structure of the model. First, we provide a pathwise Loynes' type construction which enables to prove the existence of a unique matching for the model defined over all the integers. Second, we prove that the model is dynamically reversible: we define an exchange transformation in which we interchange the positions of each matched pair, and show that the items in the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we obtain product form stationary distributions of several new Markov chains associated with the model. As a by product, we compute useful performance measures, for instance the link lengths between matched items.Comment: 33 pages, 12 figure