218 research outputs found
Sojourn time in a single server queue with threshold service rate control
We study the sojourn time in a queueing system with a single exponential
server, serving a Poisson stream of customers in order of arrival. Service is
provided at low or high rate, which can be adapted at exponential inspection
times. When the number of customers in the system is above a given threshold,
the service rate is upgraded to the high rate, and otherwise, it is downgraded
to the low rate. The state dependent changes in the service rate make the
analysis of the sojourn time a challenging problem, since the sojourn time now
also depends on future arrivals. We determine the Laplace transform of the
stationary sojourn time and describe a procedure to compute all moments as
well. First we analyze the special case of continuous inspection, where the
service rate immediately changes once the threshold is crossed. Then we extend
the analysis to random inspection times. This extension requires the
development of a new methodological tool, that is "matrix generating
functions". The power of this tool is that it can also be used to analyze
generalizations to phase-type services and inspection times.Comment: 16 pages, 13 figure
Local stability in a transient Markov chain
We prove two propositions with conditions that a system, which is described
by a transient Markov chain, will display local stability. Examples of such
systems include partly overloaded Jackson networks, partly overloaded polling
systems, and overloaded multi-server queues with skill based service, under
first come first served policy.Comment: 6 page
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
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