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1.
This paper exposes the stochastic structure of traffic processes in a class of finite state queueing systems which are modeled
in continuous time as Markov processes. The theory is presented for theM/E
k
/φ/L class under a wide range of queue disciplines. Particular traffic processes of interest include the arrival, input, output,
departure and overflow processes. Several examples are given which demonstrate that the theory unifies many earlier works,
as well as providing some new results. Several extensions to the model are discussed. 相似文献
2.
Vyacheslav M. Abramov 《Acta Appl Math》2008,100(3):201-226
The paper studies closed queueing networks containing a server station and k client stations. The server station is an infinite server queueing system, and client stations are single-server queueing
systems with autonomous service, i.e. every client station serves customers (units) only at random instants generated by a
strictly stationary and ergodic sequence of random variables. The total number of units in the network is N. The expected times between departures in client stations are (N
μ
j
)−1. After a service completion in the server station, a unit is transmitted to the jth client station with probability p
j
(j=1,2,…,k), and being processed in the jth client station, the unit returns to the server station. The network is assumed to be in a semi-Markov environment. A semi-Markov
environment is defined by a finite or countable infinite Markov chain and by sequences of independent and identically distributed
random variables. Then the routing probabilities p
j
(j=1,2,…,k) and transmission rates (which are expressed via parameters of the network) depend on a Markov state of the environment.
The paper studies the queue-length processes in client stations of this network and is aimed to the analysis of performance
measures associated with this network. The questions risen in this paper have immediate relation to quality control of complex
telecommunication networks, and the obtained results are expected to lead to the solutions to many practical problems of this
area of research.
相似文献
3.
We derive rough and exact asymptotic expressions for the stationary distribution π of a Markov chain arising in a queueing/production context. The approach we develop can also handle “cascades,” which are
situations where the fluid limit of the large deviation path from the origin to the increasingly rare event is nonlinear.
Our approach considers a process that starts at the rare event. In our production example, we can have two sequences of states
that asymptotically lie on the same line, yet π has different asymptotics on the two sequences. 相似文献
4.
We consider the stability of parallel server systems under the longest queue first (LQF) rule. We show that when the underlying
graph of a parallel server system is a tree, the standard nominal traffic condition is sufficient for the stability of that
system under LQF when interarrival and service times have general distributions. Then we consider a special parallel server
system, which is known as the X-model, whose underlying graph is not a tree. We provide additional “drift” conditions for
the stability and transience of these queueing systems with exponential interarrival and service times. Drift conditions depend
in general on the stationary distribution of an induced Markov chain that is derived from the underlying queueing system.
We illustrate our results with examples and simulation experiments. We also demonstrate that the stability of the LQF depends
on the tie-breaking rule used and that it can be unstable even under arbitrary low loads. 相似文献
5.
Vladimir V. Anisimov 《TOP》1999,7(2):169-186
Some special classes of Switching Processes such as Recurrent Processes of a Semi-Markov type and Processes with Semi-Markov
Switches are introduced. Limit theorems of Averaging Principle and Diffusion Approximation types are given. Applications to
the asymptotic analysis of overloading state-dependent Markov and semi-Markov queueing modelsM
SM,Q
/M
SM,Q
/1/∞ and retrial queueing systemsM/G/1/w.r in transient conditions are studied.
The paper was supported by INTAS Project 96-0828 相似文献
6.
M/G/1 type queueing systems whose arrival rate is a function of an independent continuous time Markov chain are considered.
We suggest a simple analytical approach which allows rigorous mathematical analysis of the stationary characteristics under
heavy traffic. Their asymptotic behaviour is described in terms of characteristics of the modulating process (defined as a
solution of a set of linear algebraic equations). The analysis is based on certain “semi-explicit” formulas for the performance
characteristics.
This research was supported by INTAS under grant No. 96-0828. 相似文献
7.
A two-station, four-class queueing network with dynamic scheduling of servers is analyzed. It is shown that the corresponding
Markov decision problem converges under fluid scaling to a fluid optimal control model. The structure of the optimal policy
for the fluid network, and of an asymptotically optimal policy for the queueing network are derived in an explicit form. They
concur with the tandem μ-rule, if this policy gives priority to the same flow of customers in both stations. In general, they
are monotone with a linear switching surface.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
8.
Summary This paper is concerned with the study of a newM/G/1 retrial queueing system in which the delays between retrials are exponentially distributed random variables with linear intensityg(n)=α+nμ, when there aren≥1 customers in the retrial group. This new retrial discipline will be calledlinear control policy. We carry out an extensive analysis of the model, including existence of stationary regime, stationary distribution of the
embedded Markov chain at epochs of service completions, joint distribution of the orbit size and the server state in steady
state and busy period. The results agree with known results for special cases. 相似文献
9.
We analyse a single-server retrial queueing system with infinite buffer, Poisson arrivals, general distribution of service
time and linear retrial policy. If an arriving customer finds the server occupied, he joins with probabilityp a retrial group (called orbit) and with complementary probabilityq a priority queue in order to be served. After the customer is served completely, he will decide either to return to the priority
queue for another service with probability ϑ or to leave the system forever with probability
=1−ϑ, where 0≤ϑ<1. We study the ergodicity of the embedded Markov chain, its stationary distribution function and the joint
generating function of the number of customers in both groups in the steady-state regime. Moreover, we obtain the generating
function of system size distribution, which generalizes the well-knownPollaczek-Khinchin formula. Also we obtain a stochastic decomposition law for our queueing system and as an application we study the asymptotic behaviour
under high rate of retrials. The results agree with known special cases. Finally, we give numerical examples to illustrate
the effect of the parameters on several performance characteristics. 相似文献
10.
S. Minkevicius 《Lithuanian Mathematical Journal》2005,45(3):299-314
The modern queueing theory is a powerful tool for a quantitative and qualitative analysis of communication systems, computer
networks, transportation systems, and many other technical systems. The paper is designated to the analysis of queueing systems
arising in the network theory and communications theory (such as the so-called multiphase queueing systems, tandem queues,
or series of queueing systems). We present heavy traffic limit theorems for the full idle time in multiphase queueing systems.
We prove functional limit theorems for values of the full idle time of a queueing system, which is its important probability
characteristic.
__________
Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 367–386, July–September, 2005. 相似文献
11.
The paper investigates the queueing process in stochastic systems with bulk input, batch state dependent service, server vacations,
and three post-vacation disciplines. The policy of leaving and entering busy periods is hysteretic, meaning that, initially,
the server leaves the system on multiple vacation trips whenever the queue falls below r (⩾1), and resumes service when during his absence the system replenishes to N or more customers upon one of his returns. During his vacation trips, the server can be called off on emergency, limiting
his trips by a specified random variable (thereby encompassing several classes of vacation queues, such as ones with multiple
and single vacations). If by then the queue has not reached another fixed threshold M (⩽ N), the server enters a so-called “post-vacation period” characterized by three different disciplines: waiting, or leaving
on multiple vacation trips with or without emergency. For all three disciplines, the probability generating functions of the
discrete and continuous time parameter queueing processes in the steady state are obtained in a closed analytic form. The
author uses a semi-regenerative approach and enhances fluctuation techniques (from his previous studies) preceding the analysis
of queueing systems. Various examples demonstrate and discuss the results obtained.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
12.
13.
We consider a discrete time single server queueing system where the service time of a customer is one slot, and the arrival
process is governed by a discrete autoregressive process of order p (DAR(p)). For this queueing system, we investigate the tail behavior of the queue size and the waiting time distributions. Specifically,
we show that if the stationary distribution of DAR(p) input has a tail of regular variation with index −β−1, then the stationary distributions of the queue size and the waiting time have tails of regular variation with index −β.
This research was supported by the MIC (Ministry of Information and Communication), Korea, under the ITRC (Information Technology
Research Center) support program supervised by the IITA (Institute of Information Technology Assessment). 相似文献
14.
One of the key performance measures in queueing systems is the decay rate of the steady-state tail probabilities of the queue
lengths. It is known that if a corresponding fluid model is stable and the stochastic primitives have finite moments, then
the queue lengths also have finite moments, so that the tail probability ℙ(⋅>s) decays faster than s
−n
for any n. It is natural to conjecture that the decay rate is in fact exponential. 相似文献
15.
Qi-Ming He 《European Journal of Operational Research》2000,120(3):641
This paper studies a multi-server queueing system with multiple types of customers and last-come-first-served (LCFS) non-preemptive service discipline. First, a quasi-birth-and-death (QBD) Markov process with a tree structure is defined and some classical results of QBD Markov processes are generalized. Second, the MMAP[K]/PH[K]/N/LCFS non-preemptive queue is introduced. Using results of the QBD Markov process with a tree structure, explicit formulas are derived and an efficient algorithm is developed for computing the stationary distribution of queue strings. Numerical examples are presented to show the impact of the correlation and the pattern of the arrival process on the queueing process of each type of customer. 相似文献
16.
Obtaining (tail) probabilities from a transform function is an important topic in queueing theory. To obtain these probabilities
in discrete-time queueing systems, we have to invert probability generating functions, since most important distributions
in discrete-time queueing systems can be determined in the form of probability generating functions. In this paper, we calculate
the tail probabilities of two particular random variables in discrete-time priority queueing systems, by means of the dominant
singularity approximation. We show that obtaining these tail probabilities can be a complex task, and that the obtained tail
probabilities are not necessarily exponential (as in most ‘traditional’ queueing systems). Further, we show the impact and
significance of the various system parameters on the type of tail behavior. Finally, we compare our approximation results
with simulations. 相似文献
17.
We propose a new queueing model named the acquisition queue. It differs from conventional queueing models in that the server
not only serves customers, but also performs acquisition of new customers. The server has to divide its energy between both
activities. The number of newly acquired customers is uncertain, and the effect of the server’s acquisition efforts can only
be seen after some fixed time period δ (delay).
The acquisition queue constitutes a (δ+1)-dimensional Markov chain. The limiting queue length distribution is derived in terms of its probability generating function,
and an exact expression for the mean queue length is given. For large values of δ the numerical procedures needed for calculating the mean queue length become computationally cumbersome. We therefore complement
the exact expression with a fluid approximation.
One of the key features of the acquisition queue is that the server performs more acquisition when the queue is small. Together
with the delay, this causes the queue length process to show a strongly cyclic behavior. We propose and investigate several
ways of planning the acquisition efforts. In particular, we propose an acquisition scheme that is designed specifically to
reduce the cyclic behavior of the queue length process.
This research was financially supported by the European Network of Excellence Euro-NGI. The work of the second author was
supported in part by a TALENT grant from the Netherlands Organisation for Scientific Research (NWO). 相似文献
18.
The asymptotic behavior of a queueing process in overloaded state-dependent queueing models (systems and networks) of a switching structure is investigated. A new approach to study fluid and diffusion approximation type theorems (without reflection) in transient and quasi-stationary regimes is suggested. The approach is based on functional limit theorems of averaging principle and diffusion approximation types for so-called Switching processes. Some classes of state-dependent Markov and non-Markov overloaded queueing systems and networks with different types of calls, batch arrival and service, unreliable servers, networks (M
SM,Q
/M
SM,Q
/1/)
r
switched by a semi-Markov environment and state-dependent polling systems are considered. 相似文献
19.
Fabrice Guillemin Bruno Sericola 《Methodology and Computing in Applied Probability》2007,9(4):521-540
Motivated by queueing systems playing a key role in the performance evaluation of telecommunication networks, we analyze in
this paper the stationary behavior of a fluid queue, when the instantaneous input rate is driven by a continuous-time Markov
chain with finite or infinite state space. In the case of an infinite state space and for particular classes of Markov chains
with a countable state space, such as quasi birth and death processes or Markov chains of the G/M/1 type, we develop an algorithm
to compute the stationary probability distribution function of the buffer level in the fluid queue. This algorithm relies
on simple recurrence relations satisfied by key characteristics of an auxiliary queueing system with normalized input rates.
相似文献
20.
We study the limit behavior of the emptying times of anM
r
/G/1/n single-channel queueing system under heavy load conditions. It is assumed that the arrival stream is governed by a Markov chain admitting state consolidation.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 55–61, 1986. 相似文献