Second part of the Special Issue on Product Forms, Stochastic Matching, and Redundancy

Queueing Systems: Theory and Applications dedicated to product forms, stochastic matching, and redundancy models.In the analysis of queueing systems, one's goal typically is to derive closed-form expressions for performance metrics of interest, e.g., the distribution of the number of jobs in the system or of response time. Often, one assumes that the system under consideration has Markovian arrival and service processes, thus enabling the use of Markov chain analysis. The vast majority of the time, however, the associated Markov chain is complicated and does not admit a straightforward analysis. Consequently, a wealth of techniques have been developed to solve intricate Markov chains, either exactly or in approximation.Of particular interest to us in this special collection of papers are processes that yield a so-called "product-form" stationary distribution. Here, we use the term "product-form" to refer to the scenario where the stationary distribution of an individual queue can be expressed as a product of terms, with one term corresponding to each job in the system. This type of product-form result came to prominence with the seminal work of Krzesinski on Order Independent (OI) queues [10, 11, 28, 29], and has been of particular interest in recent years because product forms emerge in the analysis of two important application domains: redundancy systems and stochastic matching models.In a redundancy system, when a job arrives it joins the queue at all servers with which it is compatible, where a bipartite graph specifies the job-server compatibilities. One can thus think of the job as having "copies" present in multiple servers' queues; the extra copies are immediately removed from the system when the first copy either enters service (cancel-on-start) or completes service (cancel-on-completion). The redundancy system with cancel-on-completion was shown to have a product-form stationary distribution in [24]; it was later shown that this result for redundancy systems follows directly from the product-form for OI queues [12]. In skill-based service systems-which are equivalent to cancel-on-start redundancy systems-each arriving customer is assigned to some server at which it is processed, again with permissible matchings specified by a bipartite compatibility graph. Under the so-called FCFS-ALIS (First-Come, First Served -Assign the Longest Idling Server) matching rule, such a system has a productform stationary distribution [3, 16, 39]. By stochastic matc