96 research outputs found
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
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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 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
Many Server Scaling of the N-System Under FCFS-ALIS
The N-System with independent Poisson arrivals and exponential
server-dependent service times under first come first served and assign to
longest idle server policy has explicit steady state distribution. We scale the
arrival and the number of servers simultaneously, and obtain the fluid and
central limit approximation for the steady state. This is the first step
towards exploring the many server scaling limit behavior of general parallel
service systems
Reversibility and further properties of FCFS infinite bipartite matching
The model of FCFS infinite bipartite matching was introduced in
caldentey-kaplan-weiss 2009. In this model there is a sequence of items that
are chosen i.i.d. from and an independent
sequence of items that are chosen i.i.d. from ,
and a bipartite compatibility graph between and
. Items of the two sequences are matched according to the
compatibility graph, and the matching is FCFS, each item in the one sequence is
matched to the earliest compatible unmatched item in the other sequence. In
adan-weiss 2011 a Markov chain associated with the matching was analyzed, a
condition for stability was verified, a product form stationary distribution
was derived and the rates of matches between compatible types
and were calculated.
In the current paper, we present several new results that unveil the
fundamental structure of the model. First, we provide a pathwise Loynes' type
construction which enables to prove the existence of a unique matching for the
model defined over all the integers. Second, we prove that the model is
dynamically reversible: we define an exchange transformation in which we
interchange the positions of each matched pair, and show that the items in the
resulting permuted sequences are again independent and i.i.d., and the matching
between them is FCFS in reversed time. Third, we obtain product form stationary
distributions of several new Markov chains associated with the model. As a by
product, we compute useful performance measures, for instance the link lengths
between matched items.Comment: 33 pages, 12 figure
A queue with skill based service under FCFS-ALIS : steady state, overloaded system, and behavior under abandonments
We consider a queueing system with servers S = {m_1, …, m_J}, and with customer types C = {a, b, …}. A bipartite graph G describes which pairs of server - customer type are compatible. We consider FCFS-ALIS policy: A server always picks the first, longest waiting compatible customer, a customer is always assigned to the longest idle compatible server. We assume Poisson arrivals and exponential service times. We derive an explicit product-form expression for the steady state distribution of this system when service capacity is sufficient. We analyze the system under overload, when partial steady state exists. Finally we describe the behavior of the system with generally distributed abandonments, under many arrivals - fast service scaling
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