Approximation of periodic queues

1987 ◽  
Vol 19 (3) ◽  
pp. 691-707 ◽  
Author(s):  
Tomasz Rolski
Keyword(s):  
Weak Convergence ◽  
Time Of Day ◽  
Uniform Integrability ◽  
Queue Size ◽  
Approximation Theorems ◽  
Mean Delay ◽  
Periodic Queues ◽  
Poisson Arrivals ◽  
Markov Modulated

In this paper we demonstrate how some characteristics of queues with the periodic Poisson arrivals can be approximated by the respective characteristics in queues with Markov modulated input. These Markov modulated queues were recently studied by Regterschot and de Smit (1984). The approximation theorems are given in terms of the weak convergence of some characteristics and their uniform integrability. The approximations are applicable for the following characteristics: mean workload, mean workload at the time of day, mean delay, mean queue size.

Approximation of periodic queues

1987 ◽  
Vol 19 (03) ◽  
pp. 691-707 ◽  
Author(s):  
Tomasz Rolski
Keyword(s):  
Weak Convergence ◽  
Time Of Day ◽  
Uniform Integrability ◽  
Queue Size ◽  
Approximation Theorems ◽  
Mean Delay ◽  
Periodic Queues ◽  
Poisson Arrivals ◽  
Markov Modulated

In this paper we demonstrate how some characteristics of queues with the periodic Poisson arrivals can be approximated by the respective characteristics in queues with Markov modulated input. These Markov modulated queues were recently studied by Regterschot and de Smit (1984). The approximation theorems are given in terms of the weak convergence of some characteristics and their uniform integrability. The approximations are applicable for the following characteristics: mean workload, mean workload at the time of day, mean delay, mean queue size.

scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (02) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

NEW ESTIMATORS FOR EFFICIENT GI/G/1 SIMULATION

2005 ◽  
Vol 19 (2) ◽  
pp. 219-239
Author(s):  
Chia-Li Wang ◽  
Ronald W. Wolff
Keyword(s):  
Recent Article ◽  
Idle Period ◽  
Mean Delay ◽  
Direct Spread ◽  
Poisson Arrivals ◽  
Stationary Delay

For simulating GI/G/1 queues, we investigate estimators of stationary delay-in-queue moments that were suggested but not investigated in our recent article and we develop new ones that are even more efficient. Among them are direct spread estimators that are functions of a generated sequence of spread idle periods and are combinations of estimators. We also develop corresponding conditional estimators of equilibrium idle-period moments and delay moments. We show that conditional estimators are the most efficient; in fact, for Poisson arrivals, they are exact. In simulation runs with both Erlang and hyperexponential arrivals, conditional estimators of mean delay are more efficient than a published method that estimates idle-period moments by factors well over 100 and by factors of over 800 to several thousand for estimating stationary delay variance.

scholarly journals Unimodular random trees

2013 ◽  
Vol 35 (2) ◽  
pp. 359-373 ◽  
Author(s):  
ITAI BENJAMINI ◽  
RUSSELL LYONS ◽  
ODED SCHRAMM
Keyword(s):  
Weak Convergence ◽  
Degree Distribution ◽  
Local Convergence ◽  
Hyperbolic Plane ◽  
Random Trees ◽  
Uniform Integrability ◽  
Ideal Boundary ◽  
Bounded Degree ◽  
Regular Tree ◽  
Finite Graphs

AbstractWe consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also note that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.

Weak convergence of Markov-modulated random sequences

Stochastics ◽  
2010 ◽  
Vol 82 (6) ◽  
pp. 521-552 ◽  
Author(s):  
Son Luu Nguyen ◽  
G. Yin
Keyword(s):  
Weak Convergence ◽  
Random Sequences ◽  
Markov Modulated

scholarly journals Periodic queues in heavy traffic

1989 ◽  
Vol 21 (2) ◽  
pp. 485-487 ◽  
Author(s):  
G. I. Falin
Keyword(s):  
Wiener Process ◽  
Waiting Time ◽  
Diffusion Approximation ◽  
Queueing System ◽  
Heavy Traffic ◽  
Time Process ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Periodic Queues ◽  
Poisson Arrivals

An analytic approach to the diffusion approximation in queueing due to Burman (1979) is applied to the M(t)/G/1/∞ queueing system with periodic Poisson arrivals. We show that under heavy traffic the virtual waiting time process can be approximated by a certain Wiener process with reflecting barrier at 0.

Analysis of two parallel queues by common exit service and infinite queue size

1981 ◽  
Vol 13 (1) ◽  
pp. 147-166
Author(s):  
C. Atkinson ◽  
M. E. Thompson
Keyword(s):  
Integral Equations ◽  
Service Time ◽  
Queue Size ◽  
Parallel Queues ◽  
Fixed Length ◽  
Common Gate ◽  
Erlangian Service Times ◽  
Poisson Arrivals ◽  
Service Times ◽  
Independent And Identically Distributed

A system of two parallel queues is considered, where each customer must leave after service through a common gate G. It is assumed that service times at the two stations I and II are independent and identically distributed, and that exit service takes a fixed length of time. A I-customer may be served at station I only if the previous I-customer has completed exit service. Integral equations are formulated from which the distribution of the total service time may be obtained when the two queue sizes are infinite. These equations are solved for exponential and generalized erlangian service times. Extensions to the case of k parallel queues and to the case of Poisson arrivals and finite queue sizes are discussed briefly.

Approximations for the steady-state probabilities in the M/G/c queue

1981 ◽  
Vol 13 (1) ◽  
pp. 186-206 ◽  
Author(s):  
H. C. Tijms ◽  
M. H. Van Hoorn ◽  
A. Federgruen
Keyword(s):  
Steady State ◽  
Queue Size ◽  
Numerical Experience ◽  
General Service Times ◽  
Server Queue ◽  
Poisson Arrivals ◽  
The Mean ◽  
Multi Server ◽  
General Service ◽  
Steady State Probabilities

For the multi-server queue with Poisson arrivals and general service times we present various approximations for the steady-state probabilities of the queue size. These approximations are computed from numerically stable recursion schemes which can be easily applied in practice. Numerical experience reveals that the approximations are very accurate with errors typically below 5%. For the delay probability the various approximations result either into the widely used Erlang delay probability or into a new approximation which improves in many cases the Erlang delay probability approximation. Also for the mean queue size we find a new approximation that turns out to be a good approximation for all values of the queueing parameters including the coefficient of variation of the service time.

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