Sojourn time asymptotics in processor-sharing queues

Over the past few decades, the Processor-Sharing (PS) discipline has attracted a great deal of attention in the queueing literature. While the PS paradigm emerged in the sixties as an idealization of round-robin scheduling in time-shared computer systems, it has recently captured renewed interest as a useful concept for modeling the flow-level performance of bandwidth-sharing protocols in communication networks. In contrast to the simple geometric queue length distribution, the sojourn time lacks such a nice closed-form characterization, even for exponential service requirements. In case of heavy-tailed service requirements however, there exists a simple asymptotic equivalence between the sojourn time and the service requirement distribution, which is commonly referred to as a reduced service rate approximation. In the present survey paper, we give an overview of several methods that have been developed to obtain such an asymptotic equivalence under various distributional assumptions. We outline the differences and similarities between the various approaches, discuss some connections, and present necessary and sufficient conditions for an asymptotic equivalence to hold. We also consider the generalization of the reduced service rate approximation to several extensions of the M/G/1 PS queue. In addition, we identify a relationship between the reduced service rate approximation and a queue length distribution with a geometrically decaying tail, and extend it to so-called bandwidth-sharing networks. The state-of-the-art with regard to sojourn time asymptotics in PS queues with light-tailed service requirements is also briefly described. Last, we reflect on some possible avenues for further research. AMS Subject Classification 60K25 (primary), 60F10, 68M20, 90B18, 90B22 (secondary).     

Stationary tail probabilities in exponential server tandems with renewal arrivals

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths of the arrival and virtual service processes in order to get a large queue size at the second queue. We then use large deviations estimates of the probabilities of these paths, and solve a constrained convex optimization problem to find the most likely path leading to a large queue size. We find that the stationary queue length distribution at the second queue has an exponentially decaying tail, and obtain the exact rate of decay.Research supported in part by NSF grant NCR 88-57731 and the AT & T Foundation.     

Tail asymptotics for the queue length in an M/G/1 retrial queue

In this paper, we study the tail behavior of the stationary queue length of an M/G/1 retrial queue. We show that the subexponential tail of the stationary queue length of an M/G/1 retrial queue is determined by that of the corresponding M/G/1 queue, and hence the stationary queue length in an M/G/1 retrial queue is subexponential if the stationary queue length in the corresponding M/G/1 queue is subexponential. Our results for subexponential tails also apply to regularly varying tails, and we provide the regularly varying tail asymptotics for the stationary queue length of the M/G/1 retrial queue. AMS subject classifications: 60J25, 60K25     

Decay rate for a PH/M/2 queue with shortest queue discipline

In this paper, we consider a PH/M/2 queue in which each server has its own queue and arriving customers join the shortest queue. For this model, it has been conjectured that the decay rate of the tail probabilities for the shortest queue length in the steady state is equal to the square of the decay rate for the queue length in the corresponding PH/M/2 model with a single queue. We prove this fact in the sense that the tail probabilities are asymptotically geometric when the difference of the queue sizes and the arrival phase are fixed. Our proof is based on the matrix analytic approach pioneered by Neuts and recent results on the decay rates. AMS subject classifications: 60K25 · 60K20 · 60F10 · 90B22     

Heavily loaded queue coupled to two underloaded queues

We consider a system of three parallel queues with Poisson arrivals and exponentially distributed service requirements. The service rate for the heavily loaded queue depends on which of the two underloaded queues are empty. We derive the lowest-order asymptotic approximation to the joint stationary distribution of the queue lengths, in terms of a small parameter measuring the closeness of the heavily loaded queue to instability. To this order the queue lengths are independent, and the underloaded queues and the heavily loaded queue have geometrically and, after suitable scaling, exponentially distributed lengths, respectively. The expression for the exponential decay rate for the heavily loaded queue involves the solution to an inhomogeneous linear functional equation. Explicit results are obtained for this decay rate when the two underloaded queues have vastly different arrival and service rates.     

Rare event asymptotics for a random walk in the quarter plane

This paper presents a novel technique for deriving asymptotic expressions for the occurrence of rare events for a random walk in the quarter plane. In particular, we study a tandem queue with Poisson arrivals, exponential service times and coupled processors. The service rate for one queue is only a fraction of the global service rate when the other queue is non-empty; when one queue is empty, the other queue has full service rate. The bivariate generating function of the queue lengths gives rise to a functional equation. In order to derive asymptotic expressions for large queue lengths, we combine the kernel method for functional equations with boundary value problems and singularity analysis.     

Asymptotic behaviour of the loss probability of theM/G/1/K andG/M/1/K queues

This paper investigates the asymptotic behaviour of the loss probability of theM / G/1/K and G/M/1/K queues as the buffer size increases. It is shown that the loss probability approaches its limiting value, which depends on the offered load, with an exponential decay in essentially all cases. The value of the decay rate can be easily computed from the main queue parameters. Moreover, the close relation existing between the loss behaviour of the two examined queueing systems is highlighted and a duality concept is introduced. Finally some numerical examples are given to illustrate on the usefulness of the asymptotic approximation.     

Maximum entropy analysis to the N policy M/G/1 queueing system with a removable server

We study a single removable server in an M/G/1 queueing system operating under the N policy in steady-state. The server may be turned on at arrival epochs or off at departure epochs. Using the maximum entropy principle with several well-known constraints, we develop the approximate formulae for the probability distributions of the number of customers and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with exact analytic results for three different service time distributions, exponential, 2-stage Erlang, and 2-stage hyper-exponential. The maximum entropy approximation approach is accurate enough for practical purposes. We demonstrate, through the maximum entropy principle results, that the N policy M/G/1 queueing system is sufficiently robust to the variations of service time distribution functions.     

Large deviations of the waiting time in the GI/G/1 queue with random order service

We consider the GI/G/1 queue where customers are served in random order and the service time distribution has a finite exponential moment. We derive the large deviations result for the waiting time distribution by showing that the asymptotic decay rate of the waiting time distribution is the same as that of the busy period distribution.     

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.     

Exact Asymptotics for k i -limited Exponential Polling Models

In this paper we find exact asymptotic expressions for the event that the total queue length is large for a k _i -limited exponential polling model with equal service rates and two classes of customer. It is found that this behaviour divides into two very different regimes, depending on the arrival rates to the system. Using these exact asymptotic expressions, we provide heuristics for choosing the k _i values to provide a given level of quality of service to one class while giving best effort to the other class.     

A large-deviations analysis of the GI/GI/1 SRPT queue

We consider a GI/GI/1 queue with the shortest remaining processing time discipline (SRPT) and light-tailed service times. Our interest is focused on the tail behavior of the sojourn-time distribution. We obtain a general expression for its large-deviations decay rate. The value of this decay rate critically depends on whether there is mass in the endpoint of the service-time distribution or not. An auxiliary priority queue, for which we obtain some new results, plays an important role in our analysis. We apply our SRPT results to compare SRPT with FIFO from a large-deviations point of view. 2000 Mathematics Subject Classification: Primary—60K25; Secondary—60F10; 90B22     

Impact of Bursty Traffic on Queues

The impact of bursty traffic on queues is investigated in this paper. We consider a discrete-time single server queue with an infinite storage room, that releases customers at the constant rate of c customers/slot. The queue is fed by an M/G/∞ process. The M/G/∞ process can be seen as a process resulting from the superposition of infinitely many ‘sessions’: sessions become active according to a Poisson process; a station stays active for a random time, with probability distribution G, after which it becomes inactive. The number of customers entering the queue in the time-interval [t, t + 1) is then defined as the number of active sessions at time t (t = 0,1, ...) or, equivalently, as the number of busy servers at time t in an M/G/∞ queue, thereby explaining the terminology. The M/G/∞ process enjoys several attractive features: First, it can display various forms of dependencies, the extent of which being governed by the service time distribution G. The heavier the tail of G, the more bursty the M/G/∞ process. Second, this process arises naturally in teletraffic as the limiting case for the aggregation of on/off sources [27]. Third, it has been shown to be a good model for various types of network traffic, including telnet/ftp connections [37] and variable-bit-rate (VBR) video traffic [24]. Last but not least, it is amenable to queueing analysis due to its very strong structural properties. In this paper, we compute an asymptotic lower bound for the tail distribution of the queue length. This bound suggests that the queueing delays will dramatically increase as the burstiness of the M/G/∞ input process increases. More specifically, if the tail of G is heavy, implying a bursty input process, then the tail of the queue length will also be heavy. This result is in sharp contrast with the exponential decay rate of the tail distribution of the queue length in presence of ‘non-bursty’ traffic (e.g. Poisson-like traffic). This revised version was published online in August 2006 with corrections to the Cover Date.     

Deviation matrix and asymptotic variance for GI/M/1-type Markov chains

We investigate deviation matrix for discrete-time GI/M/1-type Markov chains in terms of the matrix-analytic method, and revisit the link between deviation matrix and the asymptotic variance. Parallel results are obtained for continuous-time GI/M/1-type Markov chains based on the technique of uniformization. We conclude with A. B. Clarke＇s tandem queue as an illustrative example, and compute the asymptotic variance for the queue length for this model.     

Asymptotic behavior of the loss probability for an M/G/1/N queue with vacations

In this paper, asymptotic properties of the loss probability are considered for an M/G/1/N queue with server vacations and exhaustive service discipline, denoted by an M/G/1/N-(V, E)-queue. Exact asymptotic rates of the loss probability are obtained for the cases in which the traffic intensity is smaller than, equal to and greater than one, respectively. When the vacation time is zero, the model considered degenerates to the standard M/G/1/N queue. For this standard queueing model, our analysis provides new or extended asymptotic results for the loss probability. In terms of the duality relationship between the M/G/1/N and GI/M/1/N queues, we also provide asymptotic properties for the standard GI/M/1/N model.     

On the infinite server shortest queue problem: Non-symmetric case

We consider two parallel M/M/∞ queues. All servers in the first queue work at rate μ₁ and all in the second work at rate μ₂. A new arrival is routed to the system with the lesser number of customers. If both queues have equal occupancy, the arrival joins the first queue with probability ν₁, and the second with probability ν₂ = 1−ν₁. We analyze this model asymptotically. We assume that the arrival rate λ is large compared to the two service rates. We give several different asymptotic formulas, that apply for different ranges of the state space. The numerical accuracy of the asymptotic results is tested. AMS subject classification 60K25 60K30 34E20     

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...     

Heavy Tails in Multi-Server Queue

In this paper, the asymptotic behaviour of the distribution tail of the stationary waiting time W in the GI/GI/2 FCFS queue is studied. Under subexponential-type assumptions on the service time distribution, bounds and sharp asymptotics are given for the probability P{W > x}. We also get asymptotics for the distribution tail of a stationary two-dimensional workload vector and of a stationary queue length. These asymptotics depend heavily on the traffic load. AMS subject classification: 60K25     

Bayesian prediction inM/M/1 queues

Simple queues with Poisson input and exponential service times are considered to illustrate how well-suited Bayesian methods are used to handle the common inferential aims that appear when dealing with queue problems. The emphasis will mainly be placed on prediction; in particular, we study the predictive distribution of usual measures of effectiveness in anM/M/1 queue system, such as the number of customers in the queue and in the system, the waiting time in the queue and in the system, the length of an idle period and the length of a busy period.