Dependence among Single queues in series

Due to the analytical intractability of general tandem queues, we develop innovative methods to quantify the dependence among stations through simulation. Dependence is defined by the contribution queue time at each station, and contribution factors are developed based on the insight from Friedman's reduction method and Jackson networks. In a tandem queue, the dependence among stations can be either diffusion or blocking, and their impact depends on the positions relative to the bottlenecks. Based on these results, we show that improving the performance of the system bottleneck may not be the most effective place to reduce system cycle time. Rather than making independence assumptions, the proposed method points out a promising direction and sheds light on the insights of the dependence in practical systems.

Tandem queues constitute a fundamental structure of queueing networks. However, exact queue times in tandem queues are notoriously difficult to compute except for some special cases. Several approximation schemes that are based on mathematical assumptions that enable approximate analyses of tandem queues have been reported in the literature. This article proposes an approximation approach that is based on observed properties of the behavior of tandem queues: the intrinsic gap and intrinsic ratio. The approach exploits the nearly linear and heavy-traffic properties of the intrinsic ratio, which appear to hold in realistic production situations. The proposed approach outperforms existing approximation methods across a broad range of examined cases. It is also demonstrated that the proposed approach has the potential when applied to historical data to achieve accurate mean queue time estimates in practical production environments. [Supplemental materials are available for this article. Go to the publisher's online edition of IIE Transactions to view the supplemental file.]

Preface. 1. Simulations and the Monte Carlo method. 1.1 Description of the method. 1.2 Convergence theorems. 1.3 Simulation of random variables. 1.4 Variance reduction techniques. 1.5 Exercises. 2. Markov chains. 2.1 Definitions and elementary properties. 2.2 Examples. 2.3 Strong Markov property. 2.4 Recurrent and transient states. 2.5 The irreducible and recurrent case. 2.6 The aperiodic case. 2.7 Reversible Markov chain. 2.8 Rate of convergence to equilibrium. 2.9 Statistics of Markov chains. 2.10 Exercises. 3. Stochastic algorithms. 3.1 Markov chain Monte Carlo. 3.2 Simulation of the invariant probability. 3.3 Rate of convergence towards the invariant probability. 3.4 Simulated annealing. 3.5 Exercises. 4. Markov chains and the genome. 4.1 Reading DNA. 4.2 The i.i.d. model. 4.3 The Markov model. 4.4 Hidden Markov models. 4.5 Hidden semi-Markov model. 4.6 Alignment of two sequences. 4.7 A multiple alignment algorithm. 4.8 Exercises. 5. Control and filtering of Markov chains. 5.1 Deterministic optimal control. 5.2 Control of Markov chains. 5.3 Linear quadratic optimal control. 5.4 Filtering of Markov chains. 5.5 The Kalman-Bucy filter. 5.6 Linear-quadratic control with partial observation. 5.7 Exercises. 6. The Poisson process. 6.1 Point processes and counting processes. 6.2 The Poisson process. 6.3 The Markov property. 6.4 Large time behaviour. 6.5 Exercises. 7. Jump Markov processes. 7.1 General facts. 7.2 Infinitesimal generator. 7.3 The strong Markov property. 7.4 Embedded Markov chain. 7.5 Recurrent and transient states. 7.6 The irreducible recurrent case. 7.7 Reversibility. 7.8 Markov models of evolution and phylogeny. 7.9 Application to discretized partial differential equations. 7.10 Simulated annealing. 7.11 Exercises. 8. Queues and networks. 8.1 M/M/1 queue. 8.2 M/M/1/K queue. 8.3 M/M/s queue. 8.4 M/M/s/s queue. 8.5 Repair shop. 8.6 Queues in series. 8.7 M/G/ queue. 8.8 M/G/1 queue. 8.9 Open Jackson network. 8.10 Closed Jackson network. 8.11 Telephone network. 8.12 Kelly networks. 8.13 Exercises. 9. Introduction to mathematical finance. 9.1 Fundamental concepts. 9.2 European options in the discrete model. 9.3 The Black-Scholes model and formula. 9.4 American options in the discrete model. 9.5 American options in the Black-Scholes model. 9.6 Interest rate and bonds. 9.7 Exercises. 10. Solutions to selected exercises. 10.1 Chapter 1. 10.2 Chapter 2. 10.3 Chapter 3. 10.4 Chapter 4. 10.5 Chapter 5. 10.6 Chapter 6. 10.7 Chapter 7. 10.8 Chapter 8. 10.9 Chapter 9. References Index.

This paper considers a queueing system consisting of two single-server queues in series, in which the service times of an arbitrary customer at both queues are identical. Customers arrive at the first queue according to a Poisson process. Of this model, which is of importance in modern network design, a rather complete analysis will be given. The results include necessary and sufficient conditions for stationarity of the tandem system, expressions for the joint stationary distributions of the actual waiting times at both queues and of the virtual waiting times at both queues, and explicit expressions (i.e., not in transform form) for the stationary distributions of the sojourn times and of the actual and virtual waiting times at the second queue. In Part II (pp. 644–659) these results will be used to obtain asymptotic and numerical results, which will provide more insight into the general phenomenon of tandem queueing with correlated service times at the consecutive queues.

Approximating class-departure variability in tandem queues with downtime events: Regression-based variability function

New bounds for expected cycle times in tandem queues with blocking

The assignment of extra servers to stations in Tandem Queueing systems with small or no buffers

We propose an enhancement to the parametric-decomposition method for calculating approximate steady-state performance measures of open queueing networks with non-Poisson arrival processes and nonexponential service-time distributions. Instead of using a variability parameter ca2 for each arrival process, we suggest using a variability function ca2ρ, 0 < ρ < 1, for each arrival process i.e, the variability parameter should be regarded as a function of the traffic intensity ρ of a queue to which the arrival process might go. Variability functions provide a convenient representation of different levels of variability in different time scales for arrival processes that are not nearly renewal processes. Variability functions enable the approximations to account for long-range effects in queueing networks that cannot be addressed by variability parameters. For example, the variability functions provide a way to address the heavy-traffic bottleneck phenomenon, in which exceptional variability (either high or low) in the input has little impact in a series of queues with low-to-moderate traffic intensities, and then has a big impact when it reaches a later queue with a relatively high traffic intensity. The variability functions also enable the approximations to characterize irregular periodic deterministic external arrival processes in a reasonable way; i.e., if there are no batches, then ca2ρ should be 0 for ρ near 0 or 1, but ca2ρ can assume arbitrarily large values for appropriate intermediate ρ. We present a full network algorithm with variability functions, showing that the idea is implementable. We also show how simulations of single queues can be effectively exploited to determine variability functions for difficult external arrival processes.

The queueing analysis of store and forward data communication networks is complicated by the fact that messages preserve their length as they traverse the system: in the queueing model service times of a message at successive queues (channels) are dependent. The present study considers the case of two communication channels in series with identical capacities and with a finite intermediate buffer. That part of a message for which there is no room in the buffer is lost. The resulting queueing model of two queues in series with identical service requirements at both queues is extensively analysed. Results include the distribution of the response time in the second queue (time in buffer plus transmission time in second channel) and of the total amount of work in the second queue. A comparison is made with the analogous queueing model with independent service requirements.

Obtaining exact blocking probabilities for tandem queues with finite capacities is not a trivial problem. In this paper, we propose a computational approximation method using max-plus algebra for computing blocking probability in a Poisson-driven 2-node tandem queue with finite capacities and constant service times. The blocking probability of a finite-capacity queueing system can be obtained from either the tail probability of stationary waiting time or the difference between two expected stationary waiting times at the first node of the corresponding extended 3-node tandem queue. The computational results in this study show that the proposed approach provides a good approximation of the blocking probability, and in particular, it works well under moderately to heavily loaded situations. The proposed approach is not limited to a particular blocking policy, system structure, or service time; hence, it is applicable to general queues with finite buffer capacities and various blocking policies.

Using a definition of partial ordering of distribution functions, it is proven that for a tandem queueing system with many stations in series, where each station can have either one server with an arbitrary service distribution or a number of constant servers in parallel, the expected total waiting time in system of every customer decreases as the interarrival and service distributions becomes smaller with respect to that ordering. Some stronger conclusions are also given under stronger order relations. Using these results, bounds for the expected total waiting time in system are then readily obtained for wide classes of tandem queues.

The equilibrium behavior of Jackson queueing networks (Poisson arrivals, exponential servers and Bernoulli switches) has recently been investigated in some detail. In particular, it was found that in equilibrium, the traffic processes on the so-called exit arcs of a Jackson network with single server nodes constitute Poisson processes—a result extending Burke's theorem from single queues to networks of queues. A conjecture made by Burke and others contends that the traffic processes on non-exit arcs cannot be Poisson in equilibrium. This paper proves this conjecture to be true for a variety of Jackson networks with single server nodes. Subsequently, a number of characterizations of the equilibrium traffic streams on the arcs of open Jackson networks emerge, whereby Poisson-related stochastic properties of traffic streams are shown to be equivalent to a simple graph-theoretical property of the underlying arcs. These results then help to identify some inherent limitations on the feasibility of equilibrium decompositions of Jackson networks, and to point out conditions under which further decompositions are ‘approximately’ valid.

Dependence among servers is the root cause of the analytic intractability of general queueing networks. A tandem queue is the smallest unit possessing the dependence. Understand its behavior is the key to understand the behavior of general queueing networks. Based on observed properties, such as intrinsic gap and intrinsic ratio, a new approximation approach for tandem queues is proposed. Across a broad range of examined cases, this new approach outperforms prior methods based on stochastic independence or diffusion approximations.

Impact of Behavioral Factors on Performance of Multi-Server Queueing Systems

Tandem queues are good mathematical models for description of information transmission in various communication systems and networks. These queues play also an important role for the validation of different decomposition algorithms designed for investigating more general queueing networks. So, their investigation is interesting for theory and applications. In this paper, we consider tandem queue suitable for modeling the systems and networks where information flows are correlated and bursty what is typical for many modern telecommunication networks. Possible correlation of customers inter-arrival times and batch arrivals are taken into account via of consideration of the Batch Markovian Arrival Process (BMAP) as input stream to the system. The system consists of two stations. The service time at the station 1 is assumed to be generally distributed. There is no buffer at this station, and customers who meet the busy server repeat attempts to enter the system in random time intervals. The service process at the station 2 is assumed to be described by the continuous time Markov chain with a finite state space. This assumption holds good, e.g., if the station 2 has a finite buffer, consists of a finite number of identical or heterogeneous servers where the service time distribution is assumed to be of PH (PHase) type. Markov chain embedded at service completion epochs at the station 1 and the process of system states at arbitrary time are under study. Ergodicity condition and algorithms for computing the steady state probabilities are presented.