Heavy-traffic analysis for the GI/G/1 queue with heavy-tailed distributions

We consider a GI/G/1 queue in which the service time distribution and/or the interarrival time distribution has a heavy tail, i.e., a tail behaviour like t ^−ν with 1 < ν ⩽ 2 , so that the mean is finite but the variance is infinite. We prove a heavy-traffic limit theorem for the distribution of the stationary actual waiting time W. If the tail of the service time distribution is heavier than that of the interarrival time distribution, and the traffic load a → 1, then W, multiplied by an appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the Kovalenko distribution. If the tail of the interarrival time distribution is heavier than that of the service time distribution, and the traffic load a → 1, then W, multiplied by another appropriate ‘coefficient of contraction’ that is a function of a, converges in distribution to the negative exponential distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

Conditioned limit theorem for virtual waiting time process of the GI/G/1 queue

We prove that in the queueing system GI/G/1 with traffic intensity one, the virtual waiting time process suitably scaled, normed and conditioned by the event that the length of the first busy period exceeds n converges to the Brownian meander process, as n .     

Approximations for the conditional waiting times in the GI/G/c queue

This note presents a two-moment approximation for the conditional average waiting time in the standard multi-server queue and an approximation for the tail probabilities of the conditional waiting time distribution in the standard single-server queue. These approximations have been tested by extensive numerical experiments.     

Approximations for waiting time in GI/G/1 systems

The Sokolov procedure is described and used to obtain an explicit and easily applied approximation for the waiting time distribution in the FIFO GI/G/1 queue.     

Rejection rules in theM/G/1 queue

We consider aM/G/1 queue modified such that an arriving customer may be totally or partially rejected depending on a r.v. (the barricade) describing his impatience and on the state of the system. Three main variants of this scheme are studied. The steady-state distribution is expressed in terms of Volterra equations and the relation to storage processes, dams and queues with state-dependent Poisson arrival rate is discussed. For exponential service times, we further find the busy period Laplace transform in the case of a deterministic barricade, whereas for exponential barricade it is shown by a coupling argument that the busy period can be identified with a first passage time in an associated birth-death process.     

The MacLaurin expansion for a G/G/1 queue with Markov-modulated arrivals and services

Consider a Markov-modulated G/G/1 queueing system in which the arrival and the service mechanisms are controlled by an underlying Markov chain. The classical approaches to the waiting time of this type of queueing system have severe computational difficulties. In this paper, we develop a numerical algorithm to calculate the moments of the waiting time based on Gong and Hu's idea. Our numerical results show that the algorithm is powerful. A matrix recursive equation for the moments of the waiting time is also given under certain conditions.     

Some new results on waiting time and busy time in M/G/1 queue

This paper considers an M/G/1 queue with Poisson rate λ > 0 and service time distribution G(t) which is supposed to have finite mean 1/μ. The following questions are first studied: (a) The closed bounds of the probability that waiting time is more than a fixed value; (b)The total busy time of the server, which including the distribution, probability that are more than a fixed value during a given time interval (0, t], and the expected value. Some new and important results are obtained by theories of the classes of life distributions and renewal process.     

Fluid approximation and its convergence Rate for GI/G/1 queue with vacations

A GI/G/1 queue with vacations is considered in this paper.We develop an approximating technique on max function of independent and identically distributed(i.i.d.) random variables,that is max{ηi,1 ≤ i ≤ n}.The approximating technique is used to obtain the fluid approximation for the queue length,workload and busy time processes.Furthermore,under uniform topology,if the scaled arrival process and the scaled service process converge to the corresponding fluid processes with an exponential rate,we prove by the...     

Tail probabilities of low-priority waiting times and queue lengths in MAP/GI/1 queues

We consider the problem of estimating tail probabilities of waiting times in statistical multiplexing systems with two classes of sources – one with high priority and the other with low priority. The priority discipline is assumed to be nonpreemptive. Exact expressions for the transforms of these quantities are derived assuming that packet or cell streams are generated by Markovian Arrival Processes (MAPs). Then a numerical investigation of the large-buffer asymptotic behavior of the the waiting-time distribution for low-priority sources shows that these asymptotics are often non-exponential.     

Combined Elapsed Time and Matrix-Analytic Method for the Discrete Time GI/G/1 and GI X /G/1 Systems

In this paper, we show that the discrete GI/G/1 system can be easily analysed as a QBD process with infinite blocks by using the elapsed time approach in conjunction with the Matrix-geometric approach. The positive recurrence of the resulting Markov chain is more easily established when compared with the remaining time approach. The G-measure associated with this Markov chain has a special structure which is usefully exploited. Most importantly, we show that this approach can be extended to the analysis of the GI ^X /G/1 system. We also obtain the distributions of the queue length, busy period and waiting times under the FIFO rule. Exact results, based on computational approach, are obtained for the cases of input parameters with finite support – these situations are more commonly encountered in practical problems.     

Central limit theorems for a branching random walk with a random environment in time

We consider a branching random walk with a random environment in time, in which the offspring distribution of a particle of generation n and the distribution of the displacements of its children depend on an environment indexed by the time n. The environment is supposed to be independent and identically distributed. For A ?, let Z_n(A) be the number of particles of generation n located in A. We show central limit theorems for the counting measure Z_n(·) with appropriate normalization.     

两类负顾客M/GI/1系统的统计平衡条件

负顾客排队模型由于其灵活模拟各种复杂随机现象的广阔的应用前景，当前正越来越受到各类高性能通讯网络研究多方面的广泛关注．由于负顾客的抵消作用这类系统可以容许在顾客到达率大于服务率的情况下，进入平稳状态．本文用马尔可夫更新理论和Foster负偏移准则，研究了两类M／GI／1负顾客排队模型进入平稳状态的充要条件，首次得到了负顾客更新到达情况下，带负顾客抵消队列头部正顾客和队列尾部正顾客两种策略下的M／GI／1（FCFS）系统的统计平衡条件．当负顾客到达取更新过程的特例一泊松过程时，这一结果与Harrison＆Pital（1996）中所得结果完全一致．     

Operating characteristics of MX/G/1 queue with N-policy

We consider aM ^X/G/1 queueing system withN-policy. The server is turned off as soon as the system empties. When the queue length reaches or exceeds a predetermined valueN (threshold), the server is turned on and begins to serve the customers. We place our emphasis on understanding the operational characteristics of the queueing system. One of our findings is that the system size is the sum of two independent random variables: one has thePGF of the stationary system size of theM ^X/G/1 queueing system withoutN-policy and the other one has the probability generating function _j=0 ^N=1 _j z ^j/ _j=0 ^N=1 _j , in which _j is the probability that the system state stays atj before reaching or exceedingN during an idle period. Using this interpretation of the system size distribution, we determine the optimal thresholdN under a linear cost structure.     

基于最大熵的延迟启动/关闭N策略M/G/1可修排队稳态队长分布

研究具有延迟启动-关闭的N策略M/G/1可修排队系统,利用最大熵方法导出稳态队长分布的解析解,进一步得到基于最大熵的顾客平均等待时间.通过比较顾客的平均等待时间来检验最大熵方法的精度,结果表明基于最大熵方法得到的稳态队长分布是相当精确的.     

Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time

We consider anM/G/1 queue with FCFS queue discipline. We present asymptotic expansions for tail probabilities of the stationary waiting time when the service time distribution is longtailed and we discuss an extension of our methods to theM ^[x]/G/1 queue with batch arrivals.     

Performance analysis of a GI/M/1 queue with single working vacation

Consider a GI/M/1 queue with single working vacation. During the vacation period, the server works at a lower rate rather than stopping completely, and only takes one vacation each time. Using the matrix analytic approach, the steady-state distributions of the number of customers in the system at both arrival and arbitrary epochs are obtained. Then the closed property of the conditional probability of gamma distribution is proved and using it the waiting time of an arbitrary customer is analyzed. Finally, Some numerical results and effect of critical model parameters on performance measures have been presented.     

The M/G/c queue in light traffic

Under light traffic, we investigate the quality of a well‐known approximation for first‐moment performance measures for an M/G/c queue, and, in particular, conditions under which the approximation is either an upper or a lower bound. The approach is to combine known relationships between quantities such as average delay and time‐average work in system with direct sample‐path comparisons of system operation under two modes of operation: conventional FIFO and a version of preemptive LIFO. We then use light traffic limit theorems to show an inequality between time‐average work of the M/G/c queue and that of the approximation. In the process, we obtain new and improved approximations. This revised version was published online in June 2006 with corrections to the Cover Date.     

Representing workloads in GI/G/1 queues through the preemptive-resume LIFO queue discipline

We give in this paper a detailed sample-average analysis of GI/G/1 queues with the preemptive-resume LIFO (last-in-first-out) queue discipline: we study the long-run state behavior of the system by averaging over arrival epochs, departure epochs, as well as time, and obtain relations that express the resulting averages in terms of basic characteristics within busy cycles. These relations, together with the fact that the preemptive-resume LIFO queue discipline is work-conserving, imply new representations for both actual and virtual delays in standard GI/G/1 queues with the FIFO (first-in-first-out) queue discipline. The arguments by which our results are obtained unveil the underlying structural explanations for many classical and somewhat mysterious results relating to queue lengths and/or delays in standard GI/G/1 queues, including the well-known Bene's formula for the delay distribution in M/G/l. We also discuss how to extend our results to settings more general than GI/G/1.     

On stochastic decomposition in the GI/M/1 queue with single exponential vacation

We consider a GI/M/1 queueing system in which the server takes exactly one exponential vacation each time the system empties. We derive the PGF of the stationary queue length and the LST of the stationary FIFO sojourn time. We show that both the queue length and the sojourn time can be stochastically decomposed into meaningful quantities.     

The conditional sojourn time distribution in the GI/M/1 processor-sharing queue in heavy traffic

We treat the GI/M/1 queue with a processor-sharing server, in the heavy traffic case. Using perturbation methods, we construct asymptotic expansions for the conditional sojourn time distribution of a tagged customer conditioned on the tagged customer's service time. The resulting approximation is simple in form and involves only the first three moments of the interarrival time distribution.