Sojourn time distributions in the queue defined by a general QBD process

We consider a general QBD process as defining a FIFO queue and obtain the stationary distribution of the sojourn time of a customer in that queue as a matrix exponential distribution, which is identical to a phase-type distribution under a certain condition. Since QBD processes include many queueing models where the arrival and service process are dependent, these results form a substantial generalization of analogous results reported in the literature for queues such as the PH/PH/c queue. We also discuss asymptotic properties of the sojourn time distribution through its matrix exponential form.     

Approximate sojourn time distribution of a discriminatory processor sharing queue with impatient customers

We consider a two-class processor sharing queueing system scheduled by the discriminatory processor sharing discipline. Poisson arrivals of customers and exponential amounts of service requirements are assumed. At any moment of being served, a customer can leave the system without completion of its service. In the asymptotic regime, where the ratio of the time scales of the two-class customers is infinite, we obtain the conditional sojourn time distribution of each class customers. Numerical experiments show that the time scale decomposition approach developed in this paper gives a good approximation to the conditional sojourn time distribution as well as the expectation of it.     

Asymptotic expansions for the sojourn time distribution in the M/G/1-PS queue

We consider the M/G/1 queue with a processor sharing server. We study the conditional sojourn time distribution, conditioned on the customer’s service requirement, as well as the unconditional distribution, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. Our results demonstrate the possible tail behaviors of the unconditional distribution, which was previously known in the cases G = M and G = D (where it is purely exponential). We assume that the service density decays at least exponentially fast. We use various methods for the asymptotic expansion of integrals, such as the Laplace and saddle point methods.     

Joint distribution of sojourn time and queue length in the M/G/1 queue with (in) finite capacity

For the M/G/1 queue we study the joint distribution of the number of customers x present immediately before an arrival epoch and of the residual service time ζ of the customer in service at this epoch. The correlation coefficient ? (x, ζ) is shown to be positive (negative) when the service time distribution is DFR (IFR). The result for the joint distribution of x and ζ leads to the joint distribution of x, of the sojourn time s of the arriving customer and of the number of customers z left behind by this customer at his departure. ?(x, s), ?(z, s) and ?(x, z) are shown to be positive; ?(x, s) and ?(z, s) are compared in some detail.Subsequently the M/G/1 queue with finite capacity is considered; the joint distributions of x and ζ and of x and s are derived. These results may be used to study the cycle time distribution in a two-stage cyclic queue.     

A semi-Markov process with an inverse Gaussian distribution as sojourn time

Let X_n denote the state of a device after n repairs. We assume that the time between two repairs is the time τ taken by a Wiener process {W(t), t ? 0}, starting from w₀ and with drift μ < 0, to reach c ∈ [0, w₀). After the nth repair, the process takes on either the value X_n?1 + 1 or X_n?1 + 2. The probability that X_n = X_n?1 + j, for j = 1, 2, depends on whether τ ? t₀ (a fixed constant) or τ > t₀. The device is considered to be worn out when X_n ? k, where k ∈ {1, 2, …}. This model is based on the ones proposed by Rishel (1991) [1] and Tseng and Peng (2007) [2]. We obtain an explicit expression for the mean lifetime of the device. Numerical methods are used to illustrate the analytical findings.     

Insensitive bounds for the moments of the sojourn time distribution in the M/G/1 processor-sharing queue

This paper studies the M/G/1 processor-sharing (PS) queue, in particular the sojourn time distribution conditioned on the initial job size. Although several expressions for the Laplace-Stieltjes transform (LST) are known, these expressions are not suitable for computational purposes. This paper derives readily applicable insensitive bounds for all moments of the conditional sojourn time distribution. The instantaneous sojourn time, i.e., the sojourn time of an infinitesimally small job, leads to insensitive upper bounds requiring only knowledge of the traffic intensity and the initial job size. Interestingly, the upper bounds involve polynomials with so-called Eulerian numbers as coefficients. In addition, stochastic ordering and moment ordering results for the sojourn time distribution are obtained. AMS Subject Classification: 60K25, 60E15 This work has been partially funded by the Dutch Ministry of Economic Affairs under the program ‘Technologische Samenwerking ICT-doorbraakprojecten’, project TSIT1025 BEYOND 3G.     

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.     

Asymptotics of waiting time distributions in the accumulating priority queue

Queueing Systems - We analyze the asymptotics of waiting time distributions in the two-class accumulating priority queue with general service times. The accumulating priority queue was suggested by...     

基于特征根方法的M/G_N/1个性化服务排队顾客逗留时间分布函数的数值计算

以多语种便民服务热线为实际应用背景,研究个性化服务M/G_N/1排队系统中顾客逗留时间分布函数的数值计算方法.首先,利用嵌入Markov链技术和Pollaczek-Khintchine变换公式给出顾客逗留时间的Laplace-Stieltjes(LS)变换.其次,根据个性化服务时间分布函数的具体类型,给出上述LS变换的有理函数表达形式.通过求解有理函数分母之具有负实部的零点,即所谓的特征根,最终使用部分分式分解方法和复分析中的留数理论给出顾客逗留时间的概率分布函数.     

Locality of reference and the use of sojourn time variance for measuring queue unfairness

Sojourn time variance is widely used as an indication of queue unfairness. We demonstrate that this quantity has a disadvantage, since it is not local to the busy period in which it is measured. We show that RAQFM, a recently proposed queue fairness metric, does possess such a locality property.     

Asymptotic behavior of the stationary distribution in a finite QBD process with zero mean drift

We consider a finite QBD process with m levels. Assuming that the mean drift is 0, we obtain an asymptotic behavior as m→∞ in the stationary distribution , by finding an explicit expression for vector c. This solves the problem that was conjectured by Miyazawa et al. [Asymptotic behaviors of the loss probability for a finite buffer queue with QBD structure, 23 (2007) 79-95].     

Asymptotics for the stationary distribution in a discrete-time two-dimensional quasi-birth-and-death process

We consider a discrete-time two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ on $\mathbb{Z}_{+}^{2}$ with a background process {J _n } on a finite set, where individual processes $\{L_{n}^{(1)}\}$ and $\{L_{n}^{(2)}\}$ are both skip free. We assume that the joint process $\{Y_{n}\}=\{(L_{n}^{(1)},L_{n}^{(2)},J_{n})\}$ is Markovian and that the transition probabilities of the two-dimensional process $\{(L_{n}^{(1)},L_{n}^{(2)})\}$ are modulated depending on the state of the background process {J _n }. This modulation is space homogeneous, but the transition probabilities in the inside of $\mathbb{Z}_{+}^{2}$ and those around the boundary faces may be different. We call this process a discrete-time two-dimensional quasi-birth-and-death (2D-QBD) process, and obtain the decay rates of the stationary distribution in the coordinate directions. We also distinguish the case where the stationary distribution asymptotically decays in the exact geometric form, in the coordinate directions.     

Asymptotics for orthogonal polynomials defined by a recurrence relation

Asymptotic expansions are given for orthogonal polynomials when the coefficients in the three-term recursion formula generating the orthogonal polynomials form sequences of bounded variation.     

On sojourn times in the finite capacity queue with processor sharing

We consider a processor shared M/M/1 queue that can accommodate at most a finite number K of customers. We give an exact expression for the sojourn time distribution in this finite capacity model, in terms of a Laplace transform. We then give the tail behavior, for the limit K→∞.     

Exact asymptotic formulae of the stationary distribution of a discrete-time two-dimensional QBD process

We propose an analytically tractable approach for studying the transient behavior of multi-server queueing systems and feed-forward networks. We model the queueing primitives via polyhedral uncertainty sets inspired by the limit laws of probability. These uncertainty sets are characterized by variability parameters that control the degree of conservatism of the model. Assuming the inter-arrival and service times belong to such uncertainty sets, we obtain closed-form expressions for the worst case transient system time in multi-server queues and feed-forward networks with deterministic routing. These analytic formulas offer rich qualitative insights on the dependence of the system times as a function of the variability parameters and the fundamental quantities in the queueing system. To approximate the average behavior, we treat the variability parameters as random variables and infer their density by using ideas from queues in heavy traffic under reflected Brownian motion. We then average the worst case values obtained with respect to the variability parameters. Our averaging approach yields approximations that match the diffusion approximations for a single queue with light-tailed primitives and allows us to extend the framework to heavy-tailed feed-forward networks. Our methodology achieves significant computational tractability and provides accurate approximations for the expected system time relative to simulated values.     

Heavy traffic limit theorems for a queue with Poisson ON/OFF long-range dependent sources and general service time distribution

In Internet environment, traffic flow to a link is typically modeled by superposition of ON/OFF based sources. During each ON-period for a particular source, packets arrive according to a Poisson process and packet sizes (hence service times) can be generally distributed. In this paper, we establish heavy traffic limit theorems to provide suitable approximations for the system under first-in first-out (FIFO) and work-conserving service discipline, which state that, when the lengths of both ON- and OFF-periods are lightly tailed, the sequences of the scaled queue length and workload processes converge weakly to short-range dependent reflecting Gaussian processes, and when the lengths of ON- and/or OFF-periods are heavily tailed with infinite variance, the sequences converge weakly to either reflecting fractional Brownian motions (FBMs) or certain type of longrange dependent reflecting Gaussian processes depending on the choice of scaling as the number of superposed sources tends to infinity. Moreover, the sequences exhibit a state space collapse-like property when the number of sources is large enough, which is a kind of extension of the well-known Little??s law for M/M/1 queueing system. Theory to justify the approximations is based on appropriate heavy traffic conditions which essentially mean that the service rate closely approaches the arrival rate when the number of input sources tends to infinity.     

The CLT for Markov chains with a countable state space embedded in the space lp

We find necessary and sufficient conditions for the CLT for Markov chains with a countable state space embedded in the space l_p for p⩾1. This result is an extension of the uniform CLT over the family of indicator functions in Levental (Stochastic Processes Appl. 34 (1990) 245–253), where the result is equivalent to our case p=1. A similar extension for the uniform CLT over a family of possibly unbounded functions in Tsai (Taiwan. J. Math. 1(4) (1997) 481–498) is also obtained.     

Distribution of sojourn time for a Brownian motion with jumps

We consider a Brownian motion with jumps that is a sum of a Brownian motion and compound Poisson process. It is assumed that the distribution of jumps is symmetrically exponential. A formula for the Laplace transform of the distribution of time spent by a Brownian motion with jumps over some level is obtained. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 101–116.     

The waiting time distribution for a correlated queue with exponential interarrival and service times

We are concerned with the analysis of the waiting time distribution in an $M∕M∕1$ queue in which the interarrival time between the $n$th and the $(n+1)$th customers and the service time of the $n$th customer are correlated random variables with Downton’s bivariate exponential distribution. In this paper we show that the conditional waiting time distribution, given that the waiting time is positive, is exponential.