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 f… Show more
“…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.…”
Section: Motivation and Backgroundmentioning
confidence: 99%
“…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].…”
Section: Example 1: the 'N'-modelmentioning
confidence: 99%
“…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].…”
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 .
“…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.…”
Section: Motivation and Backgroundmentioning
confidence: 99%
“…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].…”
Section: Example 1: the 'N'-modelmentioning
confidence: 99%
“…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].…”
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 .
“…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.…”
Section: Introductionmentioning
confidence: 99%
“…
We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles.
We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles. There are finite sets C and S of customer and server classes, respectively. Time is discrete and at each time step one customer and one server arrive in the system according to a joint probability measure µ on C × S, independently of the past. Also, at each time step, pairs of matched customers and servers, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph (C, S, E ⊂ C ×S). A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete-time Markov chain. We study its stability under various admissible matching policies, including ML (match the longest), MS (match the shortest), FIFO (match the oldest), Random (match uniformly), and Priority. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and Priority policies, we exhibit a bipartite graph with a nonmaximal stability region.
We consider a stochastic bipartite matching model consisting of multi-class customers and multiclass servers. Compatibility constraints between the customer and server classes are described by a bipartite graph. Each time slot, exactly one customer and one server arrive. The incoming customer (resp. server) is matched with the earliest arrived server (resp. customer) with a class that is compatible with its own class, if there is any, in which case the matched customer-server couple immediately leaves the system; otherwise, the incoming customer (resp. server) waits in the system until it is matched. Contrary to classical queueing models, both customers and servers may have to wait, so that their roles are interchangeable. While (the process underlying) this model was already known to have a product-form stationary distribution, this paper derives a new compact and manageable expression for the normalization constant of this distribution, as well as for the waiting probability and mean waiting time of customers and servers. We also provide a numerical example and make some important observations.
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