Fcfs infinite bipartite matching of servers and customers

Abstract

We consider an infinite sequence of customers of types C = {1, 2, . . . , I } and an infinite sequence of servers of types S = {1, 2, . . . , J }, where a server of type j can serve a subset of customer types C(j ) and where a customer of type i can be served by a subset of server types S(i). We assume that the types of customers and servers in the infinite sequences are random, independent, and identically distributed, and that customers and servers are matched according to their order in the sequence, on a first-come-first-served (FCFS) basis. We investigate this process of infinite bipartite matching. In particular, we are interested in the rate r i,j that customers of type i are assigned to servers of type j . We present a countable state Markov chain to describe this process, and for some previously unsolved instances, we prove ergodicity and existence of limiting rates, and calculate r i,j .

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

“…Both models are closely related to the model of this paper. 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].…”

“…We state the infinite matching model of [4] here (see also [18]): There are two infinite sequences, a sequence of customers c 1 , c 2 , . .…”

Fluid and diffusion approximations of probabilistic matching systems

“…They also study stability of some simple systems. Adan and Weiss [1] prove that the conditions conjectured in [5] are necessary and sufficient and they prove that the stationary probabilities have a product form. Bušić et al [4] generalize this model by dropping the independence of arriving types.…”

“…For these models, once the types of users are known, the matchings occur deterministically and the main goal is to devise policies to decide on which users should be matched with each other. Caldentey et al [5] introduce a matching system with two classes of users, namely customers and servers, where each class has several types. The types of servers with which a customer of a given type can match are determined using a bipartite graph.…”

“…They consider matching policies other than first-come-first-served. Using the fact that the conditions conjectured in [5] are necessary for stability, they show that matching the longest queue has a maximal stability region, i.e., the system is stable for any probability measure that satisfies the necessary conditions. They also show that matching the shortest and some priority policies does not have a maximal stability region and prove some sufficient conditions for the stability of these matching systems.…”