32 research outputs found
Approximate performance analysis of generalized join the shortest queue routing
In this paper we propose a highly accurate approximate performance analysis
of a heterogeneous server system with a processor sharing service discipline
and a general job-size distribution under a generalized join the shortest queue
(GJSQ) routing protocol. The GJSQ routing protocol is a natural extension of
the well-known join the shortest queue routing policy that takes into account
the non-identical service rates in addition to the number of jobs at each
server. The performance metrics that are of interest here are the equilibrium
distribution and the mean and standard deviation of the number of jobs at each
server. We show that the latter metrics are near-insensitive to the job-size
distribution using simulation experiments. By applying a single queue
approximation we model each server as a single server queue with a
state-dependent arrival process, independent of other servers in the system,
and derive the distribution of the number of jobs at the server. These
state-dependent arrival rates are intended to capture the inherent correlation
between servers in the original system and behave in a rather atypical way.Comment: 16 pages, 5 figures -- version 2 incorporates minor textual change
A Lindley-type equation arising from a carousel problem
In this paper we consider a system with two carousels operated by one picker.
The items to be picked are randomly located on the carousels and the pick times
follow a phase-type distribution. The picker alternates between the two
carousels, picking one item at a time. Important performance characteristics
are the waiting time of the picker and the throughput of the two carousels. The
waiting time of the picker satisfies an equation very similar to Lindley's
equation for the waiting time in the PH/U/1 queue. Although the latter equation
has no simple solution, we show that the one for the waiting time of the picker
can be solved explicitly. Furthermore, it is well known that the mean waiting
time in the PH/U/1 queue depends on to the complete interarrival time
distribution, but numerical results show that, for the carousel system, the
mean waiting time and throughput are rather insensitive to the pick-time
distribution.Comment: 10 pages, 1 figure, 19 reference
Analytic properties of two-carousel systems
We present analytic results for warehouse systems involving pairs of
carousels. Specifically, for various picking strategies, we show that the
sojourn time of the picker satisfies an integral equation that is a contraction
mapping. As a result, numerical approximations for performance measures such as
the throughput of the system are extremely accurate and converge fast (e.g.\
within 5 iterations) to their real values. We present simulation results
validating our results and examining more complicated strategies for pairs of
carousels.Comment: 28 pages, 17 figure
On the accuracy of phase-type approximations of heavy-tailed risk models
Numerical evaluation of ruin probabilities in the classical risk model is an
important problem. If claim sizes are heavy-tailed, then such evaluations are
challenging. To overcome this, an attractive way is to approximate the claim
sizes with a phase-type distribution. What is not clear though is how many
phases are enough in order to achieve a specific accuracy in the approximation
of the ruin probability. The goals of this paper are to investigate the number
of phases required so that we can achieve a pre-specified accuracy for the ruin
probability and to provide error bounds. Also, in the special case of a
completely monotone claim size distribution we develop an algorithm to estimate
the ruin probability by approximating the excess claim size distribution with a
hyperexponential one. Finally, we compare our approximation with the heavy
traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference
Corrected phase-type approximations of heavy-tailed queueing models in a Markovian environment
Significant correlations between arrivals of load-generating events make the
numerical evaluation of the workload of a system a challenging problem. In this
paper, we construct highly accurate approximations of the workload distribution
of the MAP/G/1 queue that capture the tail behavior of the exact workload
distribution and provide a bounded relative error. Motivated by statistical
analysis, we consider the service times as a mixture of a phase-type and a
heavy-tailed distribution. With the aid of perturbation analysis, we derive our
approximations as a sum of the workload distribution of the MAP/PH/1 queue and
a heavy-tailed component that depends on the perturbation parameter. We refer
to our approximations as corrected phase-type approximations, and we exhibit
their performance with a numerical study.Comment: Received the Marcel Neuts Student Paper Award at the 8th
International Conference on Matrix Analytic Methods in Stochastic Models 201
An infinite-server queue influenced by a semi-Markovian environment
We consider an infinite-server queue, where the arrival and service rates are both governed by a semi-Markov process that's independent of all other aspects of the queue. In particular, we derive a system of equations that are satisfied by various "parts" of the generating function of the steady-state queue-length, while assuming that all arrivals bring an amount of work to the system that's either Erlang or hyperexponentially distributed. These equations are then used to show how to derive all moments of the steady-state queue-length. We then conclude by showing how these results can be slightly extended, and used, along with a transient version of Little's law, to generate rigorous approximations of the steady-state queue length in the case that the amount of work brought by a given arrival is of an arbitrary distribution
Steady-state analysis of shortest expected delay routing
We consider a queueing system consisting of two non-identical exponential
servers, where each server has its own dedicated queue and serves the customers
in that queue FCFS. Customers arrive according to a Poisson process and join
the queue promising the shortest expected delay, which is a natural and
near-optimal policy for systems with non-identical servers. This system can be
modeled as an inhomogeneous random walk in the quadrant. By stretching the
boundaries of the compensation approach we prove that the equilibrium
distribution of this random walk can be expressed as a series of product-forms
that can be determined recursively. The resulting series expression is directly
amenable for numerical calculations and it also provides insight in the
asymptotic behavior of the equilibrium probabilities as one of the state
coordinates tends to infinity.Comment: 41 pages, 13 figure