Product‐Form Solutions for Integrated Services Packet Networks and Cloud Computing Systems

This article provides a literature survey on results for the G/G/s multiserver queue with general interarrival and service time distributions. It further presents two fundamental aspects in a concise yet relatively self‐sufficient form. These are results on the waiting time vectors, derived by Kiefer and Wolfowitz, and the special case of the G/M/s queue with exponential service time distributions, first analyzed by Kendall.

We focus on correlated queues where service and inter-arrival times are strongly correlated, and the input and output link speeds are the same. We present an analytical method to derive the LST (Laplace Stieltjes transform) of the (actual) waiting time distribution. To investigate the effect of correlation between service and inter-arrival times on the system performance, we consider a counterpart GI/G/1 queue where inter-arrival time and service time distributions are the same as in our correlated system, but they are independent. Some numerical examples are provided to show that such correlations have significant impact on the system performance.

Analytic approximations are proposed for the mean response-times of R(≥ 2) priority classes in a stable G/G/c/PR queue with general class interarrival and service time distributions and c(≥ 2) parallel servers under pre-emptive resume (PR) scheduling. The generalized exponential (GE) distributional model is used to represent general distributions with known first two moments per class. The analysis is based on the extension of known heuristic arguments and earlier results regarding the study of the stable GE/GE/c/FCFS (c ≥ 1, single class) and GE/G/1/PR queues. Numerical examples illustrate the accuracy of the proposed approximations in relation to simulations involving different interarrival and service time distributions per class. Moreover, GE-type performance bounds on the system response time per class are defined. Comments on the role of the new mean response time expressions towards the approximation of the joint and marginal queue length distributions of a stable G/G/c/PR queue are included.

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.

A E n¦ M ¦ 1 queue is defined as a single server queue with exponential service time and general cyclic arrival distributions of cycle length n. The waiting time distribution for such a queue is proved to be a sum of n exponential terms; this is a generalization of G ¦ M ¦ 1 queue results. Based on this a method for obtaining the steady-state waiting time distributions for E n¦ M ¦ 1 queues is introduced. An example is presented to show an application of E n¦ M ¦ 1 queues in deterministic routing.

We apply supervised learning to a general problem in queueing theory: using a neural net, we develop a fast and accurate predictor of the stationary system-length distribution of a GI/GI/1 queue—a fundamental queueing model for which no analytical solutions are available. To this end, we must overcome three main challenges: (i) generating a large library of training instances that cover a wide range of arbitrary interarrival and service time distributions, (ii) labeling the training instances, and (iii) providing continuous arrival and service distributions as inputs to the neural net. To overcome (i), we develop an algorithm to sample phase-type interarrival and service time distributions with complex transition structures. We demonstrate that our distribution-generating algorithm indeed covers a wide range of possible positive-valued distributions. For (ii), we label our training instances via quasi-birth-and-death(QBD) that was used to approximate PH/PH/1 (with phase-type arrival and service process) as labels for the training data. For (iii), we find that using only the first five moments of both the interarrival and service times distribution as inputs is sufficient to train the neural net. Our empirical results show that our neural model can estimate the stationary behavior of the GI/GI/1—far exceeding other available methods in terms of both accuracy and runtimes. History: Ram Ramesh, Area Editor for Data Science and Machine Learning. Funding: O. Baron received financial support from the Natural Sciences and Engineering Research Council of Canada (NERC) [Grant 458051]. D. Krass received financial support from the NERC [Grant 458395]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0263 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0263 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We obtain the sojourn time probability distribution function at equilibrium for a Markov modulated, multi-server, single queue with generalised exponential (GE) service time distribution and compound Poisson arrivals of both positive and negative customers. Such arrival processes can model both burstiness and correlated traffic and are well suited to models of ATM and other telecommunication networks. Negative customers remove (ordinary) customers in the queue and are similarly correlated and bursty. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, where a customer currently being served can and cannot be killed by a negative customer. These cases can model an unreliable server or load balancing respectively. The results are obtained as Laplace transforms and can be inverted numerically. The MM CPP/GE/c G-Queue therefore holds the promise of being a viable building block for the analysis of queues and queueing networks with bursty, correlated traffic, incorporating load balancing and node-failures, since the equilibrium behaviour of both queue lengths and response times can be determined in a tractable way.

The stability of open queueing networks

This paper examines a continuous-time routing system with general interarrival and service time distributions, operating under either the join-the-shortest-queue policy or the power-of-two-choices policy. Under a weaker set of assumptions than those commonly established in the literature, we prove that the scaled steady-state queue length at each station converges weakly to a common exponential random variable in heavy traffic. Specifically, our results hold under the assumption of the (2 + ε)th moment for the interarrival and service distributions with some ε > 0. The proof leverages the Palm version of the basic adjoint relationship (BAR) as a key technique.

Although semi-open queuing networks (SOQNs) more accurately represent many problems encountered in manufacturing and logistical applications, they have not been as widely studied as open or closed queuing networks. The SOQN is a more realistic model of a system in many instances because it includes the time a customer waits outside the system. In this paper, we provide solutions for various single-class, SOQN models assuming multiple servers and general interarrival and service time distributions. Experimental results are provided to demonstrate the effectiveness of our approach.

A method for modeling a spatial priority mechanism called the push-out buffer management policy is proposed. It is a known mechanism which takes into account two kinds of cells: vital and ordinary ones. It also ensures a high degree of flexibility for asynchronous transfer mode (ATM) services. The use of the diffusion approximation allows modeling general interarrival and service time distributions. The developed transient solution helps describe the burstiness of the flow of cells arriving at a node. The model may be useful for dimensioning network buffers within ATM networks.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

In this paper,we analyze the single-server G/G/1 queue. Let η denote the elapsed interarrival time at time 0.Assume that η≥0,we obtain the transient distribution of queue length by using the Markov skeleton processes theory.

We consider the stability of N-model systems that consist of two customer classes and two server pools. Servers in one of the pools can serve both classes, but those in the other pool can serve only one of the classes. The standard fluid models in general are not sufficient to establish the stability region of these systems under static priority policies. Therefore, we use a novel and a general approach to augment the fluid model equations based on induced Markov chains. Using this new approach, we establish the stability region of these systems under a static priority rule with thresholds when the service and interarrival times have phase-type distributions. We show that, in certain cases, the stability region depends on the distributions of the service and interarrival times (beyond their mean), on the number of servers in the system, and on the threshold value. We also show that it is possible to expand the stability region in these systems by increasing the variability of the service times (without changing their mean) while keeping the other parameters fixed. The extension of our results to parallel server systems and general service time distributions is also discussed.

The paper deals with time‐series analysis of one‐line traffic measurements in loss systems. Discrete‐time stochastic models are derived for finite‐capacity trunk groups with non‐exponential distributions of interarrival and service times. The modelling accuracy is characterized by a prediction error. Simulation results are used to assert situations when the application of the proposed modelling technique is practical. Finally the adaptive Kalman filter is used to deal with cases when the statistics of observed processes are not known a priori.