Jackson Networks Research Articles

Probability distribution in varying-structure queueing networks

The paper is devoted to the computation of stationary distributions of queueing systems in random media. Results obtained for considered models follow from a theorem proved in the paper. As an application of the theorem, we consider Jackson networks whose structure (the set of working nodes, service and input flow intensities, routing matrix, state set) and type (open/closed network) varies as the state of another network or of the environment changes. Product-form formulas for the computation of stationary distributions of the considered networks are obtained, and algorithms for the solution of auxiliary problems are developed.

Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

Service Competition with General Queueing Facilities

In many service industries, companies compete with each other on the basis of the waiting time their customers' experience, along with the price they charge for their service. A firm's waiting time standard may either be defined in terms of the expected value or a given, for instance 95%, percentile of the steady state waiting time distribution. We investigate how a service industry's competitive behavior depends on the characteristics of the service providers' queueing systems. We provide a unifying approach to investigate various standard single stage systems covering the spectrum from M/M/1 to general G/GI/s systems, along with open Jackson networks to represent multi-stage service systems. Assuming that the capacity cost is proportional with the service rates we refer to its dependence on (i) the firm's demand rate and (ii) the waiting time standard as the capacity cost function. We show that across the above road spectrum of queueing models, the capacity cost function belongs to a specific four parameter class of function, either exactly or as a close approximation. We then characterize how this capacity cost function impacts on the equilibrium behavior in the industry. We give separate treatments to the case where the firms compete in terms of (i) prices (only) (ii) their service level or waiting time standard (only), (iii) simultaneously in terms of both prices and service levels. The firms' demand rates are given by a general system of equations of the prices and waiting time standards in the industry.

Validity of heavy traffic steady-state approximations in generalized Jackson networks

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavy-traffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called “interchange-of-limits” for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

Convergence rates in monotone separable stochastic networks

We study bounds on the rate of convergence to the stationary distribution in monotone separable networks which are represented in terms of stochastic recursive sequences. Monotonicity properties of this subclass of Markov chains allow us to formulate conditions in terms of marginal network characteristics. Two particular examples, generalized Jackson networks and multiserver queues, are considered.

Product Theorem in a Jackson Network with Switching Centers

1. Classical Jackson Product Theorem A network G is called a classical Jackson network if it has Poisson input flow with intensity λ and m multi-server centers. Suppose that the ith center has ri ≤ ∞ servers (each server has exponentially distributed serving times with intensity μi) and allows an infinite queue, i = 1, 2, . . . ,m. Take an external source as a center with index 0. Determine a customer route in the network G by the route matrix Θ = (θij)i,j=0, where θij is the probability of transiting from the ith serving center to the jth serving center, θ00 = 0. Suppose that the route matrix is indivisible: ∀i, j ∈ {0, 1, . . . ,m} ∃i1, i2, . . . , ir ∈ {0, 1, ...,m}: θii1 > 0, θi1i2 > 0, . . . , θirj > 0. Thus, there is a single solution (λ, λ1, λ2, . . . , λm) of the system λ = λ, (λ, λ1, λ2, . . . , λm) = (λ, λ1, λ2, . . . , λm)Θ. Separate in the network G the ith serving center and consider it as an isolated ri-server queuing system with Poisson input flow which has intensity λi. Describe the number of customers in this system by a birth-and-death process ni(t). If ρi = λi riμi < 1, (1)

Efficient simulation of buffer overflow probabilities in jackson networks with feedback

Consider a Jackson network that allows feedback and that has a single server at each queue. The queues in this network are classified as a single ‘target’ queue and the remaining ‘feeder’ queues. In this setting we develop the large deviations limit and an asymptotically efficient importance sampling estimator for the probability that the target queue overflows during its busy period, under some regularity conditions on the feeder queue-length distribution at the initiation of the target queue busy period. This importance sampling distribution is obtained as a solution to a non-linear program. We especially focus on the case where the feeder queues, at the initiation of the target queue busy period, have the steady state distribution corresponding to these instants. In this setting, we explicitly identify the importance sampling distribution when the feeder queue service rates exceed a specified threshold. We also relate our work to the existing large deviations literature to develop a perspective on successes and limitations of our results.

Jackson networks with unlimited supply of work

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

Jackson networks with unlimited supply of work

We consider a Jackson network in which some of the nodes have an infinite supply of work: when all the customers queued at such a node have departed, the node will process a customer from this supply. Such nodes will be processing jobs all the time, so they will be fully utilized and experience a traffic intensity of 1. We calculate flow rates for such networks, obtain conditions for stability, and investigate the stationary distributions. Standard nodes in this network continue to have product-form distributions, while nodes with an infinite supply of work have geometric marginal distributions and Poisson inflows and outflows, but their joint distribution is not of product form.

Tails in generalized Jackson networks with subexponential service-time distributions

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

Tails in generalized Jackson networks with subexponential service-time distributions

We give the exact asymptotics of the tail of the stationary maximal dater in generalized Jackson networks with subexponential service times. This maximal dater, which is an analogue of the workload in an isolated queue, gives the time taken to clear all customers present at some time t when stopping all arrivals that take place later than t. We use the property that a large deviation of the maximal dater is caused by a single large service time at a single station at some time in the distant past of t, in conjunction with fluid limits of generalized Jackson networks, to derive the relevant asymptotics in closed form.

A TWO-NODE JACKSON NETWORK WITH INFINITE SUPPLY OF WORK

We consider a Jackson network with two nodes, with no exogenous input, but instead an infinite supply of work at each of the nodes: Whenever a node is empty, it processes a job from this infinite supply. We obtain an explicit expression for the steady state distribution of this system, as an infinite sum of product forms.

On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

On the Effect of Finite Buffer Truncation in a Two-Node Jackson Network

We consider a two-node Jackson network in which the buffer of node 1 is truncated. Our interest is in the limit of the tail decay rate of the queue-length distribution of node 2 when the buffer size of node 1 goes to infinity, provided that the stability condition of the unlimited network is satisfied. We show that there can be three different cases for the limit. This generalizes some recent results obtained for the tandem Jackson network. Special cases and some numerical examples are also presented.

Large deviations of a modified Jackson network: Stability and rough asymptotics

Consider a modified, stable, two node Jackson network where server 2 helps server 1 when server 2 is idle. The probability of a large deviation of the number of customers at node one can be calculated using the flat boundary theory of Schwartz and Weiss [Large Deviations Performance Analysis (1994), Chapman and Hall, New York]. Surprisingly, however, these calculations show that the proportion of time spent on the boundary, where server 2 is idle, may be zero. This is in sharp contrast to the unmodified Jackson network which spends a nonzero proportion of time on this boundary.

Bridges and networks: Exact asymptotics

We extend the Markov additive methodology developed in [Ann. Appl. Probab. 9 (1999) 110–145, Ann. Appl. Probab. 11 (2001) 596–607] to obtain the sharp asymptotics of the steady state probability of a queueing network when one of the nodes gets large. We focus on a new phenomenon we call a bridge. The bridge cases occur when the Markovian part of the twisted Markov additive process is one null recurrent or one transient, while the jitter cases treated in [Ann. Appl. Probab. 9 (1999) 110–145, Ann. Appl. Probab. 11 (2001) 596–607] occur when the Markovian part is (one) positive recurrent. The asymptotics of the steady state is an exponential times a polynomial term in the bridge case, but is purely exponential in the jitter case. We apply this theory to a modified, stable, two node Jackson network where server two helps server one when server two is idle. We derive the sharp asymptotics of the steady state distribution of the number of customers queued at each node as the number of customers queued at the server one grows large. In so doing we get an intuitive understanding of the companion paper [Ann. Appl. Probab. 15 (2005) 519–541] which gives a large deviation analysis of this problem using the flat boundary theory in the book by Shwartz and Weiss. Unlike the (unscaled) large deviation path of a Jackson network which jitters along the boundary, the unscaled large deviation path of the modified network tries to avoid the boundary where server two helps server one (and forms a bridge). In the fluid limit this bridge does collapse to a straight line, but the proportion of time spent on the flat boundary tends to zero. This bridge phenomenon is ubiquitous. We also treated the bathroom problem described in the Shwartz and Weiss book and found the bridge case is present. Here we derive the sharp asymptotics of the steady state of the bridge case and we obtain the results consistent with those obtained in [SIAM J. Appl. Math. (1984) 44 1041–1053] using complex variable methods.

Modeling parallel and distributed systems with finite workloads

In studying or designing parallel and distributed systems one should have available a robust analytical model that includes the major parameters that determine the system performance. Jackson networks have been very successful in modeling computer systems. However, the ability of Jackson networks to predict performance with system changes remains an open question, since they do not apply to systems where there are population size constraints. Also, the product-form solution of Jackson networks assumes steady-state and exponential service centers or certain specialized queueing discipline. In this paper, we present a transient model for Jackson networks that is applicable to any population size and any finite workload (no new arrivals). Using several non-exponential distributions we show to what extent the exponential distribution can be used to approximate other distributions and transient systems with finite workloads. When the number of tasks to be executed is large enough, the model approaches the product-form solution (steady-state solution). We also, study the case where the non-exponential servers have queueing (Jackson networks cannot be applied). Finally, we show how to use the model to analyze the performance of parallel and distributed systems.

Local conditions for the stochastic comparison of particle systems

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.

Local conditions for the stochastic comparison of particle systems

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.