FCFS Parallel Service Systems and Matching Models

We consider three parallel service models in which customers of several types are served by several types of servers subject to a bipartite compatibility graph, and the service policy is first come first served. Two of the models have a fixed set of servers. The first is a queueing model in which arriving customers are assigned to the longest idling compatible server if available, or else queue up in a single queue, and servers that become available pick the longest waiting compatible customer, as studied by Adan and Weiss, 2014. The second is a redundancy service model where arriving customers split into copies that queue up at all the compatible servers, and are served in each queue on FCFS basis, and leave the system when the first copy completes service, as studied by Gardner et al., 2016. The third model is a matching queueing model with a random stream of arriving servers. Arriving customers queue in a single queue and arriving servers match with the first compatible customer and leave immediately with the customer, or they leave without a customer. The last model is relevant to organ transplants, to housing assignments, to adoptions and many other situations. We study the relations between these models, and show that they are closely related to the FCFS infinite bipartite matching model, in which two infinite sequences of customers and servers of several types are matched FCFS according to a bipartite compatibility graph, as studied by Adan et al., 2017. We also introduce a directed bipartite matching model in which we embed the queueing systems. This leads to a generalization of Burke's theorem to parallel service systems

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

We consider a queueing system with servers S={m1,...,mJ}, and with customer types C={a,b,...}. A bipartite graph G describes which pairs of server-customer types are compatible. We consider FCFS-ALIS policy: A server always picks the first, longest waiting compatible customer, and a customer is always assigned to the longest idle compatible server. We assume Poisson arrivals and server dependent exponential service times. We derive an explicit product-form expression for the stationary distribution of this system when service capacity is sufficient. We also calculate fluid limits of the system under overload, to show that local steady state exists. We distinguish the case of complete resource pooling when all the customers are served at the same rate by the pooled servers, and the case when the system has a unique decomposition into subsets of customer types, each of which is served at its own rate by a pooled subset of the servers. Finally, we discuss possible behavior of the system with generally distributed abandonments, under many server scaling. This paper complements and extends previous results of Kaplan, Caldentey and Weiss [18], and of Whitt and Talreja [34], as well as previous results of the authors [4, 35] on this topic. Keywords: Service systems, multi type customers, multi type skill based servers, matching of infinite sequences, product form solution, first come first served policy, assign longest idle server policy, complete resource pooling, local steady state, overloaded queues, abandonment

Approximation and Control of Skill Based Parallel Service Systems with Homogeneous Service

A skill base parallel service system is comprised of a set of customers of different classes that arrive randomly for service, a set of servers that serve those customers and a set of qualifications that defines which customer classes can be served by which server. Systems of this kind appear in a wide range of applications from the assignment of jobs to employees with different skills to network traffic routing. Literature regarding these systems has almost exclusively been focused on the asymptotic heavy traffic regime. The reason being that such an asymptotic regime is convenient to analyze and allows the derivation of exact results. However, although many applications can be well approximated by an asymptotic regime, many others can not. In this work we are especially concerned with large scale sparse systems where, despite the system being large of scale, each customer class can only be served by a small subset of the servers. After laying foundations for the model in Chapter 1 and exploring structural properties in Chapter 2 we go on to present the two main contributions of this work. In Chapter 3 we develop a set of approximations that compile to a , first of its kind, approximation scheme of matching rates of skill based parallel service system operating under the \textit{first-come-first-serve} or \textit{longest-queue-first} policies. The accuracy of the approximation is verified with extensive simulation experiments where it is shown to provide matching rate estimates with an absolute error of $3\%-5\%$ for a wide range of traffic intensities. Later, in Chapter 4 we use insights provided by the new approximation to derive weighted versions of the \textit{first-come-first-serve} or \textit{longest-queue-first} and show, through comprehensive simulation testing, that these weighted polices dramatically reduce the waiting time of customers in congested system compared to the original unweighted versions. Finally, we extend the use of the weighted policies to systems with matching rewards and show that, by appropriate choice of weights, these policies can be used by a controller to efficiently trade-off between the rate of reward accumulation and waiting time experienced by the customer