M/G/1/MLPS compared with M/G/1/PS within service time distribution class IMRL

Multilevel processor-sharing (MLPS) disciplines were originally introduced by Kleinrock (in computer applications 1976) but they were forgotten for years. However, due to an application related to the service differentiation between short and long TCP flows in the Internet, they have recently gained new interest. In this paper we show that, if the service time distribution belongs to class IMRL, the mean delay in the M/G/1 queue is reduced when replacing the PS discipline with any MLPS discipline for which the internal disciplines belong to {FB, PS}. This is a generalization of our earlier result where we restricted ourselves to the service time distribution class DHR, which is a subset of class IMRL.     

TheM/G/1 queue with processor sharing and its relation to a feedback queue

The central model of this paper is anM/M/1 queue with a general probabilistic feedback mechanism. When a customer completes his ith service, he departs from the system with probability 1–p(i) and he cycles back with probabilityp(i). The mean service time of each customer is the same for each cycle. We determine the joint distribution of the successive sojourn times of a tagged customer at his loops through the system. Subsequently we let the mean service time at each loop shrink to zero and the feedback probabilities approach one in such a way that the mean total required service time remains constant. The behaviour of the feedback queue then approaches that of anM/G/1 processor sharing queue, different choices of the feedback probabilities leading to different service time distributions in the processor sharing model. This is exploited to analyse the sojourn time distribution in theM/G/1 queue with processor sharing.Some variants are also considered, viz., anM/M/1 feedback queue with additional customers who are always present, and anM/G/1 processor sharing queue with feedback.     

Sojourn time asymptotics in the M/G/1 processor sharing queue

We show for the M/G/1 processor sharing queue that the service time distribution is regularly varying of index -ν, ν non-integer, iff the sojourn time distribution is regularly varying of index -ν. This result is derived from a new expression for the Laplace–Stieltjes transform of the sojourn time distribution. That expression also leads to other new properties for the sojourn time distribution. We show how the moments of the sojourn time can be calculated recursively and prove that the kth moment of the sojourn time is finite iff the kth moment of the service time is finite. In addition, we give a short proof of a heavy traffic theorem for the sojourn time distribution, prove a heavy traffic theorem for the moments of the sojourn time, and study the properties of the heavy traffic limiting sojourn time distribution when the service time distribution is regularly varying. Explicit formulas and multiterm expansions are provided for the case that the service time has a Pareto distribution. This revised version was published online in June 2006 with corrections to the Cover Date.     

A note on comparing response times in the M/GI/1/FB and M/GI/1/PS queues

We compare the overall mean response time (a.k.a. sojourn time) of the processor sharing (PS) and feedback (FB) queues under an M/GI/1 system. We show that FB outperforms PS under service distributions having decreasing failure rates; whereas PS outperforms FB under service distributions having increasing failure rates.     

Sojourn time distributions for the M/M/1 queue in a Markovian environment

In this paper, we study a single server queue in which both the arrival rate and service rate depend on the state of an external Markov process (called the environment) with a finite state space. Given that the environment is in state j, the mean arrival and service rates are λ_j and μ_j respectively. For such a queue, the queue length distribution is known to be matrix geometric. In this paper, we characterize the Laplace-Stieltjes transform of the sojourn time distribution under four disciplines - last come first served preemptive resume, last come first served, processor sharing and round robin. We also discuss a potential application of this queue in the are of data communication.     

Using the M/G/1 queue under processor sharing for exact simulation of queues

In Sigman (J. Appl. Probab. 48A:209–216, 2011b), a first exact simulation algorithm was presented for the stationary distribution of customer delay for FIFO M/G/c queues in which ρ=λ/μ<1 (super stable case). The key idea involves dominated coupling from the past while using the M/G/1 queue under the processor sharing (PS) discipline as a sample-path upper bound, taking advantage of its time-reversibility properties so as to be able to simulate it backwards in time. Here, we expand upon this method and give several examples of other queueing models for which this method can be used to exactly simulate from their stationary distributions. Examples include sojourn times for single-server queues under various service disciplines, tandem queues, and multi-class networks with general routing.     

Explicit formulas for the variance of conditioned sojourn times in M/D/1-PS

For the M/D/1 processor sharing queue, explicit formulas for the coefficient of variation of the sojourn time conditioned on the service time and on the residual service times of the tasks found on arrival are derived via branching representations.     

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.     

Asymptotics for M/G/1 low-priority waiting-time tail probabilities

We consider the classical M/G/1 queue with two priority classes and the nonpreemptive and preemptive-resume disciplines. We show that the low-priority steady-state waiting-time can be expressed as a geometric random sum of i.i.d. random variables, just like the M/G/1 FIFO waiting-time distribution. We exploit this structures to determine the asymptotic behavior of the tail probabilities. Unlike the FIFO case, there is routinely a region of the parameters such that the tail probabilities have non-exponential asymptotics. This phenomenon even occurs when both service-time distributions are exponential. When non-exponential asymptotics holds, the asymptotic form tends to be determined by the non-exponential asymptotics for the high-priority busy-period distribution. We obtain asymptotic expansions for the low-priority waiting-time distribution by obtaining an asymptotic expansion for the busy-period transform from Kendall's functional equation. We identify the boundary between the exponential and non-exponential asymptotic regions. For the special cases of an exponential high-priority service-time distribution and of common general service-time distributions, we obtain convenient explicit forms for the low-priority waiting-time transform. We also establish asymptotic results for cases with long-tail service-time distributions. As with FIFO, the exponential asymptotics tend to provide excellent approximations, while the non-exponential asymptotics do not, but the asymptotic relations indicate the general form. In all cases, exact results can be obtained by numerically inverting the waiting-time transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

On reducing time spent in M/G/1 systems

In this paper we define a new ‘truncated shortest processing time’ scheduling discipline and present the first two moments of the time spent in a single server queuing system (M/G/1) with Poisson arrivals and truncated shortest processing time scheduling discipline. Also for quadratic cost functions, the mean cost of time spent in an M/G/1 system under (1) first come first served. (2) shortest processing time. (3) two level shortest processing time, (4) two class non-preemptive priority, and (5) truncated shortest processing time scheduling disciplines are compared.     

Analysis of the M/M/1 Queue with Processor Sharing via Spectral Theory

We show in this paper that the computation of the distribution of the sojourn time of an arbitrary customer in a M/M/1 with the processor sharing discipline (abbreviated to M/M/1 PS queue) can be formulated as a spectral problem for a self-adjoint operator. This approach allows us to improve the existing results for this queue in two directions. First, the orthogonal structure underlying the M/M/1 PS queue is revealed. Second, an integral representation of the distribution of the sojourn time of a customer entering the system while there are n customers in service is obtained.     

Simultaneous determination of capacities and load in parallel M/M/1 queues

In this paper, we address an analytical model to simultaneously determine the processing capacity and load assigned to each processor in a multiple processor configuration within the framework of M/M/1 queues. A queueing optimization model is formulated as a nonlinear programming problem having linear constraints. We then propose an optimization algorithm that utilizes the special structure of the problem. To illustrate the applicability of our method, a small sample system has been solved.     

M/G/1/N vacation model with varying E-limited service discipline

We define and analyze anM/G/1/N vacation model that uses a service discipline that we call theE-limited with limit variation discipline. According to this discipline, the server provides service until either the system is emptied (i.e. exhausted) or a randomly chosen limit ofl customers has been served. The server then goes on a vacation before returning to service the queue again. The queue length distribution and the Laplace-Stieltjes transforms of the waiting time, busy period and cycle time distributions are found. Further, an expression for the mean waiting time is developed. Several previously analyzed service disciplines, including Bernoulli scheduling, nonexhaustive service and limited service, are special cases of the general varying limit discipline that is analyzed in this paper.     

M/G/1重试排队系统的渐近稳定性

用算子半群理论研究了带有重试排队的M/G/1系统.通过解算子方程和预解方程,证明了0是系统算子的本征值,且为虚轴上唯一的谱点.从而得出了当时间趋于无穷时系统时间依赖解收敛于稳态解的结论.     

ANALYSIS OF M/G/1 STOCHASTIC CLEARING SYSTEMS

In this paper, we consider M/G/1 queuing systems governed by a stochastic clearing mechanism, called “disaster,” which removes all workload in the system whenever it occurs to the system. The clearing mechanism of disasters can be applied to computer systems in the presence of a virus as a clearing operation of all stored messages present in the system. We present the system size distribution and the sojourn time distribution.     

具有随机N—策略的M／G／1排队系统

本文讨论具有随机N-策略的M/G/1排队系统,采用向量Markov过程方法得到该系统有关的排队指标。上述结果可以看作是普通的和N-策略的M/G/1排队系统的推广。     

Analysis of M/G/1-Queues with Setup Times and Vacations under Six Different Service Disciplines

Single server M/G/1-queues with an infinite buffer are studied; these permit inclusion of server vacations and setup times. A service discipline determines the numbers of customers served in one cycle, that is, the time span between two vacation endings. Six service disciplines are investigated: the gated, limited, binomial, exhaustive, decrementing, and Bernoulli service disciplines. The performance of the system depends on three essential measures: the customer waiting time, the queue length, and the cycle duration. For each of the six service disciplines the distribution as well as the first and second moment of these three performance measures are computed. The results permit a detailed discussion of how the expected value of the performance measures depends on the arrival rate, the customer service time, the vacation time, and the setup time. Moreover, the six service disciplines are compared with respect to the first moments of the performance measures.     

M/G/1排队论系统的渐近稳定性

通过M/G/1算子的谱分析得到了M/G/1排队论系统的渐近稳定性.首先,将系统方程转化为某一合适Banach空间上的抽象Cauchy闻题,从而引入M/G/1算子.其次,分析了M/G/1算子的谱分布,得到了0是M/G/1算子的简单本征值且M/G/1算子的谱分布在左半平面的结果.最后,利用谱分析结果和算子半群理论得到了M/...     

M/G/1/K queues withN-policy and setup times

A steady-state analysis is given for M/G/1/K queues with combinedN-policy and setup times before service periods. The queue length distributions and the mean waiting times are obtained for the exhaustive service system, the gated service system, the E-limited service system, and the G-limited service system. Numerical examples are also provided.     

具有可利用服务员的M/G/1排队模型

本文考虑了具有可利用服务员的M／G／1有有限容量的排队模型．当工作量超过k(k是常数或者随机变量),可利用服务员参与工作,一直到工作量少于或等于k．可利用服务员的速率依赖于目前工作量．应用Level-crossing方法,获得了工作量的平稳分布．应用Kolmogorov向后微分方程方法,构造更新方程以获得忙期的Laplace变换．