Product-form Distribution Research Articles

Power-of-two sampling in redundancy systems: The impact of assignment constraints

A classical sampling strategy for load balancing policies is power-of-two, where any server pair is sampled with equal probability. This does not cover practical settings with assignment constraints which force non-uniform sampling. While intuition suggests that non-uniform sampling adversely impacts performance, this was only supported through simulations, and rigorous statements have remained elusive. Building on product-form distributions for redundancy systems, we prove the stochastic dominance of uniform sampling for a four-server system as well as arbitrary-size systems in light traffic.

Optimal hyper-scalable load balancing with a strict queue limit

Load balancing plays a critical role in efficiently dispatching jobs in parallel-server systems such as cloud networks and data centers. A fundamental challenge in the design of load balancing algorithms is to achieve an optimal trade-off between delay performance and implementation overhead (e.g. communication or memory usage). This trade-off has primarily been studied so far from the angle of the amount of overhead required to achieve asymptotically optimal performance, particularly vanishing delay in large-scale systems. In contrast, in the present paper, we focus on an arbitrarily sparse communication budget, possibly well below the minimum requirement for vanishing delay, referred to as the hyper-scalable operating region. Furthermore, jobs may only be admitted when a specific limit on the queue position of the job can be guaranteed.The centerpiece of our analysis is a universal upper bound for the achievable throughput of any dispatcher-driven algorithm for a given communication budget and queue limit. We also propose a specific hyper-scalable scheme which can operate at any given message rate and enforce any given queue limit, while allowing the server states to be captured via a closed product-form network, in which servers act as customers traversing various nodes. The product-form distribution is leveraged to prove that the bound is tight and that the proposed hyper-scalable scheme is throughput-optimal in a many-server regime given the communication and queue limit constraints. Extensive simulation experiments are conducted to illustrate the results.

Time-dependent product-form Poisson distributions for reaction networks with higher order complexes.

It is well known that stochastically modeled reaction networks that are complex balanced admit a stationary distribution that is a product of Poisson distributions. In this paper, we consider the following related question: supposing that the initial distribution of a stochastically modeled reaction network is a product of Poissons, under what conditions will the distribution remain a product of Poissons for all time? By drawing inspiration from Crispin Gardiner's "Poisson representation" for the solution to the chemical master equation, we provide a necessary and sufficient condition for such a product-form distribution to hold for all time. Interestingly, the condition is a dynamical "complex-balancing" for only those complexes that have multiplicity greater than or equal to two (i.e.the higher order complexes that yield non-linear terms to the dynamics). We term this new condition the "dynamical and restricted complex balance" condition (DR for short).

Transient detailed balance and product form for reaction networks

ABSTRACTThis paper explores the boundary of the set of reaction networks that have an exact transient (truncated) multidimensional Poisson or product-form distribution for the number of particles of different types. Motivated by the birth–death process, we introduce the notions of transient detailed balance and delay functions, and use these notions to obtain the novel transient product-form distribution in a coagulation-fragmentation process for polymers with a tree-like structure from that of the pure coagulation process.

Queue decomposition & finite closed queueing network models

Closed, finite queueing networks are applicable to many different manufacturing and service system settings. The incorporation of material handling and transportation networks in finite buffer closed queueing networks is studied. A novel queue decomposition approach using state dependent queues to capture the buffer of finite M/M/1/K queues is shown to be a viable approach for modelling these systems. Each M/M/1/K queue is replaced with a coupled state dependent queue plus an M/M/1 queue. An extended mean value analysis (MVA) algorithm is employed to demonstrate the integration of the state dependent queues for the buffers in the approach. Under certain restrictions concerning the network population, finite queueing networks with the state dependent queues acting as buffers should have a product form distribution. This paper focuses on M/M/1/K queues and their transformation while future papers will treat the multi-server case. Several different closed series (i.e. cyclic), merge, and split topological systems of finite queues are analyzed and presented.

Approximately Optimal Wireless Broadcasting

We study a wireless broadcast network, where a single source reliably communicates independent messages to multiple destinations, with the potential aid of relays and cooperation between destinations. The wireless nature of the medium is captured by the broadcast nature of transmissions as well as the superposition of transmitted signals plus independent Gaussian noise at the received signal at any radio. We propose a scheme that can achieve rate tuples within a constant gap away from the cut-set bound, where the constant is independent of channel coefficients and power constraints. First, for a deterministic broadcast network, we propose a new coding scheme, constructed by adopting a “receiver-centric” viewpoint, that uses quantize-and-forward relaying as an inner code concatenated with an outer Marton code for the induced deterministic broadcast channel. This scheme is shown to achieve the cut-set bound evaluated with product form distributions. This result is then lifted to the Gaussian network by using a deterministic network called the discrete superposition network as a formal quantization interface. This two-stage construction circumvents the difficulty involved in working with a vector nonlinear non-Gaussian broadcast channel that arises if we construct a similar scheme directly for the Gaussian network.

Product-form distributions and network algorithms (abstract only)

The "product-form" characterization of the stationary distribution make a queueing network analytically a lot more tractable. This has been the primary source of inspiration in the search for "productform" characterization. In this talk, I will discuss implications of "product-form" distributions for algorithm design by means of two examples: (i) intra-queue scheduling and (ii) inter-queue scheduling in a constrained queueing network. Near the end of the talk, by means of a novel comparison result between stationary distributions of Markov chains, I will briefly discuss notion of "approximate" product-form distributions.

On the behavior of stable subnetworks in non-ergodic networks with unreliable nodes

We consider Jackson networks with unreliable nodes, which randomly break down and are under repair for a random time. The network is described by a Markov process which encompasses the availability status and queue lengths vector. Ergodicity conditions for many related networks are available in the literature and can often be expressed as rate conditions. For (reliable) nodes in Jackson networks the overall arrival rate has to be strictly less than its service rate. If for some nodes this condition is violated, the network process is not ergodic. Nevertheless, it is known that in such a situation, especially in large networks, parts of the network (where the rate condition is fulfilled) in the long run stabilize. For standard Jackson networks without breakdown of nodes, the asymptotics of such stable subnetworks were derived by Goodman and Massey [J.B. Goodman, W.A. Massey, The non-ergodic Jackson network, Journal of Applied Probability 21 (1984) 860-869]. In this paper, we obtain the asymptotics of Jackson networks with unreliable nodes and show that the state distribution of the stable subnetworks converges to a Jackson-type product form distribution. In such networks with breakdown and repair of nodes, in general, the ergodicity condition is more involved. Because no stationary distribution for the network exists, steady-state availability and performance evaluation is not possible. We show that instead assessment of the quality of service in the long run for the stabilizing subnetwork can be done by using limiting distributions. Additionally, we prove that time averages of cumulative rewards can be approximated by state-space averages.

An approximate model for base-stock-controlled assembly systems

This study is on continuously reviewed base-stock-controlled assembly systems with Poisson demand arrivals and exponential single-server facilities used for manufacturing and assembly operations. A partially aggregated but exact queueing model is developed and approximated assuming that the state-dependent transition rates arising as a result of the partial aggregation are constant. It is shown that the steady-state probability distribution of this approximate model is a product-form distribution for the simplest case with two components making up an assembly. Based on this analytical observation, similar product-form distributions are proposed for more complex assembly systems. Comparisons with simulation and matrix-geometric solutions show that the proposed product-form steady-state distributions accurately approximate relevant performance measures with a considerable advantage in terms of the required computational effort. A greedy heuristic is devised to use approximate steady-state probabilities for optimizing design parameters like base-stock levels. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplemental resources: appendix for the proofs, additional remarks, and figures and tables for the numerical results.]

Boundary behavior and product-form stationary distributions of jump diffusions in the orthant with state-dependent reflections

In this paper we consider reflected diffusions with positive and negative jumps, constrained to lie in the nonnegative orthant of ℝ n . We allow for the drift and diffusion coefficients, as well as for the directions of reflection, to be random fields over time and space. We provide a boundary behavior characterization, generalizing known results in the nonrandom coefficients and constant directions of the reflection case. In particular, the regulator processes are related to semimartingale local times at the boundaries, and they are shown not to charge the times the process expends at the intersection of boundary faces. Using the boundary results, we extend the conditions for product-form distributions in the stationary regime to the case when the drift and diffusion coefficients, as well as the directions of reflection, are random fields over space.

Boundary behavior and product-form stationary distributions of jump diffusions in the orthant with state-dependent reflections

In this paper we consider reflected diffusions with positive and negative jumps, constrained to lie in the nonnegative orthant of ℝn. We allow for the drift and diffusion coefficients, as well as for the directions of reflection, to be random fields over time and space. We provide a boundary behavior characterization, generalizing known results in the nonrandom coefficients and constant directions of the reflection case. In particular, the regulator processes are related to semimartingale local times at the boundaries, and they are shown not to charge the times the process expends at the intersection of boundary faces. Using the boundary results, we extend the conditions for product-form distributions in the stationary regime to the case when the drift and diffusion coefficients, as well as the directions of reflection, are random fields over space.

Existence and characterization of product-form invariant distributions for state-dependent stochastic networks in the heavy-traffic diffusion limit

We consider state-dependent stochastic networks in the heavy-traffic diffusion limit represented by reflected jump-diffusions in the orthant ? + n with state-dependent reflection directions upon hitting boundary faces. Jumps are allowed in each coordinate by means of independent Poisson random measures with jump amplitudes depending on the state of the process immediately before each jump. For this class of reflected jump-diffusion processes sufficient conditions for the existence of a product-form stationary density and an ergodic characterization of the stationary distribution are provided. Moreover, such stationary density is characterized in terms of semi-martingale local times at the boundaries and it is shown to be continuous and bounded. A central role is played by a previously established semi-martingale local time representation of the regulator processes.

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.

On product-form stationary distributions for reflected diffusions with jumps in the positive orthant

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density inR+nthat satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

On product-form stationary distributions for reflected diffusions with jumps in the positive orthant

In this paper, we study the stationary distributions for reflected diffusions with jumps in the positive orthant. Under the assumption that the stationary distribution possesses a density in R + n that satisfies certain finiteness conditions, we characterize the Fokker-Planck equation. We then provide necessary and sufficient conditions for the existence of a product-form distribution for diffusions with oblique boundary reflections and jumps. To do so, we exploit a recent characterization of the boundary properties of such reflected processes. In particular, we show that the conditions generalize those for semimartingale reflecting Brownian motions and reflected Lévy processes. We provide explicit results for some models of interest.

A time-reversed representation for the tail probabilities of stationary reflected Brownian motion

We consider the exponential decay rate of the stationary tail probabilities of reflected Brownian motion X in the N-dimensional orthant R + N having drift b, covariance matrix A, and constraint matrix D. Suppose that the Skorokhod or reflection mapping associated with the matrix D is well-defined and Lipschitz continuous on the space of continuous functions. Under the stability condition D −1 b<0, it is known that the exponential decay rate has a variational representation V( x). This representation is difficult to analyze, in part because there is no analytical theory associated with it. In this paper, we obtain a new representation for V( x) in terms of a time-reversed optimal control problem. Specifically, we show that V( x) is equal to the minimum cost incurred to reach the origin when starting at the point x, where the constrained dynamics are described in terms of another constraint matrix D̄, and the cost is quadratic in the control as well as the “local time” or constraining term. The equivalence of these representations in fact holds under the milder assumption that the matrices D and D̄ satisfy what is known as the completely- S condition. We then use the time-reversed representation to identify the minimizing large deviation trajectories for a class of RBMs having product form distributions. In particular, we show that the large deviation trajectories associated with product form RBMs that approximate open single-class networks or multi-class feedforward networks do not cycle.

Geometric product form queueing networks with concurrent batch movements

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

Geometric product form queueing networks with concurrent batch movements

Queueing networks have been rather restricted in order to have product form distributions for network states. Recently, several new models have appeared and enlarged this class of product form networks. In this paper, we consider another new type of queueing network with concurrent batch movements in terms of such product form results. A joint distribution of the requested batch sizes for departures and the batch sizes of the corresponding arrivals may be arbitrary. Under a certain modification of the network and mild regularity conditions, we give necessary and sufficient conditions for the network state to have the product form distribution, which is shown to provide an upper bound for the one in the original network. It is shown that two special settings satisfy these conditions. Algorithms to calculate their stationary distributions are considered, with numerical examples.

On Quasi-Reversibility and Local Balance: An Alternative Derivation of the Product-Form Results

We are concerned with a queueing network, described by a continuous-time Markov chain, in which each node is quasi-reversible. A new class of local balance equations is derived for the Markov chain with respect to a product-form distribution, which simultaneously provides an alternative and short proof for product form results of queueing networks with customers and signals. Furthermore, if each node is internally balanced, i.e., the total arrival rate equals the total departure rate for each node, the Markov chain is locally balanced in the ordinary sense. Our arguments reveal a close relationship between quasi-reversibility and local balance, and provide further insights into how transition rates are balanced in product form queueing networks.