A Reversible Erlang Loss System With Multitype Customers and Multitype Servers

Abstract

We consider a memoryless Erlang loss system with servers = {1, …, J}, and with customer types = {1, …, I}. Servers are multitype, so that server j can serve a subset of customer types C(j). We show that the probabilities of assigning arriving customers to idle servers can be chosen in such a way that the Markov process describing the system is reversible, with a simple product form stationary distribution. Furthermore, the system is insensitive; these properties are preserved for general service time distributions.

A skill based parallel service system under FCFS-ALIS — steady state, overloads, and abandonments

“…Two motivating precursors to these papers were the paper of Talreja and Whitt [34], which introduced FCFS skill based routing in an overloaded system with abandonments, and a paper of Caldentey, Kaplan and Weiss [18], which introduced the infinite matching model. Product form results for skill based routing in a loss system were obtained in [3,4].…”

“…The proof of Theorem 2.2 is given in Appendix A.2, and it is based on the proof of a similar result for a skill based service Erlang loss system [3,5].…”

A Loss System With Skill-Based Servers Under Assign to Longest Idle Server Policy

“…In [2] the same loss system is considered under a different regime. Instead of choosing the longest idle server, an arriving customer of type i which finds a set of idle servers S will choose server j ∈ S with probability P (i, j|S).…”

“…This is different from our ALIS (assign to longest idle server) regime in two ways: first, the assignment does not depend on the order in which idle servers became available, and second, the assignment is random. Because only the set of idle servers is of interest in the system of [2], it can be described by a CTMC Y (t) with state given by the set of idle servers S. It is shown in [2] that one can choose the assignment probabilities P (i, j|S) so as to make Y (t) reversible, in which case its steady state distribution is for every subset S = {j 1 , . .…”

“…It is shown in [2] that the reversible loss system in insensitive. One can ask if the same property also holds for the loss system under ALIS regime.…”