Lester Lipsky

Long-lasting transient conditions in simulations with heavy-tailed workloads

Recent evidence suggests that some characteristics of computer and telecommunications systems may be well described using heavy tailed distributions — distributions whose tail declines like a power law, which means that the probability of extremely large observations is nonnegligible. For example, such distributions have been found to describe the lengths of bursts in network traffic and the sizes of files in some systems. As a result, system designers are increasingly interested in employing heavy-tailed distributions in simulation workloads. Unfortunately, these distributions have properties considerably different from the kinds of distributions more commonly used in simulations; these properties make simulation stability hard to achieve. In this paper we explore the difficulty of achieving stability in such simulations, using tools from the theory of stable distributions. We show that such simulations exhibit two characteristics related to stability: slow convergence to steady state, and high variability at steady state. As a result, we argue that such simulations must be treated as effectively always in a transient condition. One way to address this problem is to introduce the notion of time scale as a parameter of the simulation, and we discuss methods for simulating such systems while explicitly incorporating time scale as a parameter.

Asymptotic Behavior of Total Times For Jobs That Must Start Over If a Failure Occurs.

Many processes must complete in the presence of failures. Different systems respond to task failure in different ways. The system may resume a failed task from the failure point (or a saved checkpoint shortly before the failure point), it may give up on the task and select a replacement task from the ready queue, or it may restart the task. The behavior of systems under the first two scenarios is well documented, but the third (RESTART) has resisted detailed analysis. In this paper we derive tight asymptotic relations between the distribution of task times without failures and the total time when including failures, for any failure distribution. In particular, we show that if the task-time distribution has an unbounded support, then the total-time distribution H is always heavy tailed. Asymptotic expressions are given for the tail of H in various scenarios. The key ingredients of the analysis are the Cramer–Lundberg asymptotics for geometric sums and integral asymptotics, which in some cases are obtained ...

On the completion time distribution for tasks that must restart from the beginning if a failure occurs

For many systems, failure is so common that the design choice of how to deal with it may have a significant impact on the performance of the system. There are many specific and distinct failure recovery schemes, but they can be grouped into three broad classes: RESUME, also referred to as preemptive resume (prs), or check-pointing; REPLACE, also referred to as preemptive repeat different (prd); and RESTART, also referred to as preemptive repeat identical (pri). The following describes the three recovery schemes: (1) RESUME: when a task is fails, it knows exactly where it stops, and can continue from that point when allowed to resume; (2)REPLACE: given a task fails, then when it begins processing again, it starts with a brand new task sampled from the same task time distribution; and, (3) RESTART: When a task fails, it loses all that it had acquired to up to that point and must start anew when upon continuing later. This is distinctly different from (2) since the task must run at least as long as it did before it failed, whereas a new sample, selected at random, might run for a shorter or longer time.

Impact of aggregated, self-similar ON/OFF traffic on delay in stationary queueing models (extended version)

Abstract The impact of the now widely acknowledged self-similar property of network traffic on cell-delay in a single server queueing model is investigated. The analytic traffic model, called N -Burst, uses the superposition of N independent cell streams of ON/OFF type with power-tail distributed ON periods. Queueing-delay for such arrival processes is mainly caused by over-saturation periods, which occur when too many sources are in their ON-state. The duration of the over-saturation periods is shown to have a power-tail distribution, whose exponent β is in most scenarios different from the tail exponent of the individual ON-period. Conditions on the model parameters, for which the mean and higher moments of the delay distribution become infinite, are investigated. Since these conditions depend on traffic parameters as well as on network parameters, careful network design can alleviate the performance impact of such self-similar traffic. Finally, a characterization of truncated tails by the so-called power-tail range is developed. Based on the power-tail range of the burst-length distribution, the additional parameter maximum burst size (MBS) is introduced in the N -Burst model. An asymptotic relationship between the moments of the delay distribution and the MBS is derived and is validated by the corresponding numerical results of the analytic N -Burst/M/1 queueing model.

Solutions of M/G/1//N-type Loops with Extensions to M/G/1 and GI/M/1 Queues

Closed form solutions of the joint equilibrium distribution of queue sizes are derived for a large class of M/G/1//N queues, i.e., any closed loop of two servers in which one is exponential (but possibly load dependent), and the other has a probability density function which has a rational Laplace transform and a queueing discipline that is FCFS. The class of G/M/k//N queues are included as special cases of load-dependent servers. The solutions are presented in terms of vectors and matrices whose size depends only on the distribution of the general server and not on the number of customers in the system. Efficient algorithms are outlined, and expressions for various system measurements are presented. Depending on the relative service rates of the two servers, solutions for both the M/G/1 and GI/M/1 open queues are derived as limiting cases of the M/G/1//N system. All results are contrasted with existing formulas.

On unreliable computing systems when heavy-tails appear as a result of the recovery procedure

For some computing systems, failure is rare enough that it can be ignored. In other systems, failure is so common that how to handle it can have a significant impact on the performance of the system. There are many different recovery schemes for tasks, however, they can be classified into three broad categories: 1) Resume: when a task fails, it knows exactly where it stops and can continue at that point when allowed to resume (i.e., preemptive resume - prs); 2) Replace: when a task fails, then later when the processor continues, it begins with a brand new task (i.e., preemptive repeat different prd); and, 3) Restart: when a task fails it loses all work done to that point and must start anew upon continuing later (i.e., preemptive repeat identical - pri).In this paper, assuming a computing system is unreliable, we discuss how heavy-tail (hereafter referred to as power-tail - PT) distributions can appear in a jobs task stream given the Restart recovery procedure. This is an important consideration since it is known that power-tails can lead to unstable systems [4], We then demonstrate how to obtain performance and dependablity measures for a class of computing systems comprised of P unreliable processors and a finite number of tasks N given the above recovery procedures.

A matrix-algebraic solution to two K m servers in a loop

An explicit steady-state solution is given for any queuing loop made up of two general servers, whose distribution functions have rational Laplace transforms. The solution is in matrix geometric form over a vector space that is itself a direct or Kronecker product of the internal state spaces of the two servers. The algebraic properties of relevant entities in this space are given in an appendix. The closed-form solution yields simple recursive relations that in turn lead to an efficient algorithm for calculating various performance measures such as queue length and throughput. A computational-complexity analysis shows that the algorithm requires at least an order of magnitude less computational effort than any previously reported algorithm.

On the necessity of transient performance analysis in telecommunication networks

Most analytic models of telecommunication networks are based on steady-state methods. This paper discusses potential drawbacks of steady-state parameters, in particular of the steady-state buffer-overflow or cell-loss probability when used as quality of service (QoS) criteria. The importance of transient performance analysis is demonstrated for long-range dependent (multiplexed) ON/OFF traffic. A transient parameter pair is proposed as replacement for the steady-state overflow or loss probabilities. The in-depth discussion of the behavior of those transient parameters reveals surprising results that allow for characterization and understanding of the fluctuations that are being observed in actual network behavior under traffic loads with long-range dependent properties.

Comparison of the analytic N-burst model with other approximations to telecommunications traffic

A wide variety of traffic models are presently used to study the performance of telecommunications networks. These are shown to be limiting cases of N-burst/G/1 queues. The analytic N-burst model describes traffic as superposition of N packet streams of ON/OFF type. When using power-tail distributions for the duration of the ON periods, self-similar properties, which are critical for understanding tele-traffic, are observed. For very low intraburst packet rates, the N-burst/G/1 model reduces to an M/G/1 queue. For /spl lambda//sub p/ /spl rarr/ /spl infin/ all packets in a burst arrive simultaneously and the model becomes a bulk arrival, or M/sup (X)//G/1, queue. In the same limit, the packet-based model can be compared to a model of the burst level, an M/G/1 queue where the individual customers represent complete bursts rather than individual packets. Thus the mean system time describes the mean delay for the last packet in a burst rather than the average over all packets. The continuous flow model is also shown to be a limiting case of the N-burst model by letting the number of packets in a burst, n/sub p/, and the routers packet service rate, /spl nu/, go to infinity while holding their ratio constant. Numerical results are presented comparing the steady-state results for mean packet delay and for buffer overflow probabilities of the different analytic models. They collectively show the critical importance of the burstiness parameter. The N-burst/M/1 model with self-similar properties shows drastically changing steady-state performance for specific values of the burstiness parameter. The limiting models are incapable of describing the detailed structure of the performance in this transition region.

Study of Bursty Internet Traffic

We study the effects of bursty Internet traffic through simulations. Both short-range dependency (SRD) traffic and long-range dependency (LRD) traffic are simulated over different burst parameters. The results are collected for 10 different 24 hour simulated periods in order to study and measure day-to-day statistical fluctuation. Effects of employing different traffic admission constraints are examined. An alternative for improving network throughput and utilization is proposed. Finally, a case when arrival patterns of traffic are correlated is explored.