Alexander S. Rumyantsev

Stability criterion of a multiserver model with simultaneous service

In this paper, we study a multiserver system in which each customer requires a random number of servers simultaneously and a random but identical service time at all occupied servers. This model describes the dynamics of modern high performance clusters. Stability criteria of this system (under exponential assumptions) have been proved earlier for the two-server system only. Using a matrix-analytic approach, we obtain stability criteria of such a system under exponential assumptions and with an arbitrary number of servers.

A Regeneration-Based Estimation of High Performance Multiserver Systems

In this paper we develop a novel approach to confidence estimation of the stationary measures in high performance multiserver queueing systems. This approach is based on construction of the two processes which are, respectively, upper and lower (stochastic) bounds for the trajectories of the basic queue size process in the original system. The main feature of these envelopes is that they have classical regenerations. This approach turns out to be useful when the original process is not regenerative or regenerations occur extremely rare to be applied in estimation. We apply this approach to construct confidence intervals for the steady-state queue size in classical multiserver systems and also for the novel high performance cluster (HPC) model, a multiserver system with simultaneous service. Simulation based on a real HPC dataset shows that this approach allows to estimate stationary queue size in a reasonable time with a given accuracy.

Efficient Redundancy Techniques in Cloud and Desktop Grid Systems using MAP/G/c-type Queues

Abstract Cloud computing is continuing to prove its flexibility and versatility in helping industries and businesses as well as academia as a way of providing needed computing capacity. As an important alternative to cloud computing, desktop grids allow to utilize the idle computer resources of an enterprise/community by means of distributed computing system, providing a more secure and controllable environment with lower operational expenses. Further, both cloud computing and desktop grids are meant to optimize limited resources and at the same time to decrease the expected latency for users. The crucial parameter for optimization both in cloud computing and in desktop grids is the level of redundancy (replication) for service requests/workunits. In this paper we study the optimal replication policies by considering three variations of Fork-Join systems in the context of a multi-server queueing system with a versatile point process for the arrivals. For services we consider phase type distributions as well as shifted exponential and Weibull. We use both analytical and simulation approach in our analysis and report some interesting qualitative results.

Inequalities for Workload Process in Queues with NBU/NWU Input

We give a simple proof of the well-known property PASTA for the workload and queue size process in the queueing systems with Poisson input. The proof is based on a relation connecting the workload process at an arbitrary instants and the arrival instants of the customers and, in particular, yields famous Pollaczeck-Khintchine equality. It is then shown that this equality transforms to the corresponding inequality when the interarrival time has New-Better-than-Used (NBU) or New-Worse-than-Used (NWU) distribution. Then the inequalities for the stationary workload are extended to classical multiserver system and multiserver model with simultaneous service, describing the modern high performance cluster. The analysis is illustrated by simulation of the single-server and cluster models with Weibull interarrival time, covering NBU, NWU and Poisson inputs. The obtained results are of practical interest for Quality-of-Service estimation of modern high-performance computing systems.

Stability Analysis and Simulation of a State-Dependent Transmission Rate System

In this paper, we consider a model of communication system with state-dependent service rate. This mechanism allows to change service rate to increase the efficiency of the system. Motivation of such a system is discussed as well. Then we present the regenerative proof of the sufficient stability conditions of the system which is based on the negative drift of the workload process above a high threshold. Moreover, we describe a wireless communication system in which transmission rate is being both Markov-modulated and also queue-dependent. Simulation results are presented, which illustrate the behavior of the queue size depending on the used threshold-based service rate switching mechanism.

Stability Analysis of a MAP/M/s Cluster Model by Matrix-Analytic Method

In this paper, we study the stability conditions of the multiserver system in which each customer requires a random number of servers simultaneously and a random service time, identical at all occupied servers. We call it cluster model since it describes the dynamics of the modern multicore high performance clusters (HPC). Stability criterion of an M/M/s cluster model has been proved by the authors earlier. In this work we, again using the matrix-analytic approach, prove that the stability criterion of a more general MAP/M/s cluster model (with Markov Arrival Process) has the same form as for M/M/s system. We verify by simulation that this criterion (in an appropriate form) allows to delimit stability region of a MAP/PH/s cluster model with phase-type (PH) service time distribution. Finally, we discuss asymptotic results related to accelerated stability verification, as well as to the new method of accelerated regenerative estimation of the performance metrics.

Cost and Effect of Replication and Quorum in Desktop Grid Computing

A Split–Merge multiserver model of a Desktop Grid computing system is studied. Heavy-tailed distributions are used for service times of tasks in a system, including the Pareto distribution, which allows one to obtain some analytical results. The effects of replication and quorum parameters on the key performance measures such as response time and cost of a Desktop Grid system are studied both analytically and through simulation under a variety of scenarios for system configuration and system load. Moment properties of the workload vector, which not only highlight possible heterogeneity but also play a key role in practical applications, are derived.

Evaluating a Single-Server Queue with Asynchronous Speed Scaling

We introduce an approach to energy efficiency policies evaluation in various application fields, based on widely available technical and software tools. The approach is based on a simple client-server-type application run on a single linux-operated laptop, equipped with standard frequency scaling tools. We validate the approach by analyzing two energy efficiency policies: the celebrated hysteretic control and the randomized switching (introduced recently) in a single-server queue. Explicit analytical results are obtained by means of Matrix-Analytic method, and a simulation model based on discrete event stochastic simulation (a particular case of the generalized Kiefer–Wolfowitz stochastic recursion introduced recently) allows to obtain performance and energy estimates, when analytical results are inappropriate. The results of real-world experiments are introduced, and applicability of the approach is discussed.

On Regenerative Envelopes for Cluster Model Simulation

We continue to develop a novel approach for confidence estimation of the stationary measures in the model describing high performance multiserver queueing systems, such as high performance clusters (HPC). We call this model cluster model. This model is described by a stochastic process, and in the framework of the approach, we construct two envelopes, minorant and majorant regenerative processes for the queue size process in the original system. These envelopes have classical regenerations while the original process may not be regenerative or its regenerations happen too rare to be useful for statistical estimation. It allows to construct confidence intervals for the steady-state queue size of the cluster model. We use simulation to illustrate the applicability of the approach and give recommendations how to select the predefined parameters of the envelopes to increase the efficiency of estimation. As simulation shows, the constructed envelopes allow to estimate the mean stationary queue size in the original system with a given accuracy in an acceptable time.