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 .

Fcfs infinite bipartite matching of servers and customers

“…Interest in the queueing model arises especially in the context of call centers, where various types of customer call, and are routed to various groups of skill-based servers [2], [13]. It is then often the case that the queueing system operates in balanced heavy traffic, where the sum of the λ i s equals the sum of the µ j s. If that is the case then we expect departures at the rates λ i and service with no interruptions at the rates µ j , but this is an unstable system, which is at best null recurrent.…”

“…The frequency of c n = 1 is α, the frequency of s n = 1 is β. This is the 'N'-model in the taxonomy of [13], as depicted in Figure 3. The queueing version of this system under an FCFS policy is the one analyzed in [1].…”

“…As an illustrating example, Figure 1 depicts a bipartite graph with three customer types, two server types, and customer-to-server matching S(1) = {1}, S(2) = {1, 2}, S(3) = {2} and server-to-customer matching C(1) = {1, 2}, C(2) = {2, 3}; G consists of {(1, 1), (2, 1), (2,2), (3,2)}. This is the so-called 'W'-model in the taxonomy for routeing topologies in service networks proposed in [13].…”

Stability of the Bipartite Matching Model

“…The model was introduced by Caldentey et al [3], under an additional assumption of independence between arriving customers and servers (for all c and s, µ(c, s) = µ(c, S)µ(C, s)). In the paper, the authors conjectured that any bipartite graph has a maximal stability region for the FIFO policy [3,Conjecture 4.2], and they explicitly treated some small models. In [1], Adan and Weiss proved the conjecture in a fascinating way.…”

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We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles.

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