Queue Length Vector Research Articles

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

We establish an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case.

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

The present paper establishes an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case, providing useful insight into the system dynamics. In particular, we prove that the limiting queue length vector lives in a K-dimensional cone of which the set of spanning vectors is random in general, capturing the delicate interplay between the various job types and servers. For a broad class of systems we provide a further simplification which shows that the collection of random cones constitutes a fixed K-dimensional cone, resulting in a K-dimensional SSC. The dimension~K represents the number of critically loaded subsystems, or equivalently, capacity bottlenecks in heavy-traffic, with K=1 corresponding to conventional CRP scenarios. Our approach leverages probability generating function (PGF) expressions for Markovian systems operating under redundancy policies.

Accuracy of Estimation of the Vector of Queue Lengths for Open Jackson Networks

The accuracy of approximation of the vector of queue lengths for an open Jackson network with regenerative input flow and unreliable servers is estimated. A theorem on the accuracy of approximation of the vector of queue lengths in open Jackson networks is put forward, i.e., an estimate for the probability of deviations of the norm of the difference between the process of queue lengths and the constructed reflected Brownian motion is obtained. As a corollary, an estimate of the Wasserstein distance is given.

Accuracy of estimation of the vector of queue lengths for open Jackson networks

Работа посвящена оценке точности аппроксимации вектора длин очередей для открытой сети Джексона с регенерирующим входящим потоком и ненадежными приборами. Получена теорема о точности аппроксимации вектора длин очередей в открытых сетях Джексона, т.е. оценка вероятности отклонения нормы разности процесса длин очередей и построенного отраженного броуновского движения. В качестве следствия получена оценка расстояния Вассерштейна.

Jackson network in a random environment: strong approximation

We consider a Jackson network with regenerative input flows in which every server is subject to a random environment influence generating breakdowns and repairs. They occur in accordance with two independent sequences of i.i.d. random variables. We establish a theorem on the strong approximation of the vector of queue lengths by a reflected Brownian motion in positive orthant.

Heavy-traffic limits for Discriminatory Processor Sharing models with joint batch arrivals

We study the performance of Discriminatory Processor Sharing (DPS) systems, with exponential service times and in which batches of customers of different types may arrive simultaneously according to a Poisson process. We show that the stationary joint queue-length distribution exhibits state-space collapse in heavy traffic: as the load ρ tends to 1, the scaled joint queue-length vector (1−ρ)Q converges in distribution to the product of a deterministic vector and an exponentially distributed random variable, with known parameters. The result provides new insights into the behavior of DPS systems. It shows how the queue-length distribution depends on the system parameters, and in particular, on the simultaneity of the arrivals. The result also suggests simple and fast approximations for the tail probabilities and the moments of the queue lengths in stable DPS systems, capturing the impact of the correlation structure in the arrival processes. Numerical experiments indicate that the approximations are accurate for medium and heavily loaded systems.

Analysis of Jackson networks with infinite supply and unreliable nodes

Jackson networks are versatile models for analyzing complex networks. In this paper we study generalized Jackson networks with single-server stations, where nodes may have an infinite supply of work. We allow simultaneous breakdown of servers and consider group repair strategies. We establish the existence of a steady-state distribution of the queue-length vector at stable nodes for different types of failure regimes. In steady state the distribution of the failure/repair regime and of the queue-length vector at stable nodes decouples in a product-form way. We provide closed-form solutions for the classical performance measures such as throughput or mean sojourn time at a station.

Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.

HEAVY-TRAFFIC ANALYSIS OF A NON-PREEMPTIVE MULTI-CLASS QUEUE WITH RELATIVE PRIORITIES

We study the steady-state queue-length vector in a multi-class queue with relative priorities. Upon service completion, the probability that the next served customer is from class k is controlled by class-dependent weights. Once a customer has started service, it is served without interruption until completion. We establish a state-space collapse for the scaled queue-length vector in the heavy-traffic regime, that is, in the limit the scaled queue-length vector is distributed as the product of an exponentially distributed random variable and a deterministic vector. We observe that the scaled queue length reduces as classes with smaller mean service requirement obtain relatively larger weights. We finally show that the scaled waiting time of a class-k customer is distributed as the product of two exponentially distributed random variables.

Heavy traffic analysis of the discriminatory randomorderofservice discipline

We study the steady-state queue-length vector in a multiclass single-server queue with relative priorities. Upon service completion, the probability that the next customer to be served is from class k is controlled by class-dependent weights. Once a customer has started service, it is served without interruption until completion. This is a generalization of the random-order-of-service discipline. We investigate the system in a heavy-traffic regime. We first establish a state-space collapse for the scaled queue length vector, that is, the scaled queue length vector is in the limit the product of an exponentially distributed random variable and a deterministic vector. As a direct consequence, we obtain that the scaled number of customers in the system reduces as classes with smaller mean service requirement obtain relatively larger weights. In addition, we present the distribution of the scaled sojourn time of a customer given its class, in heavy traffic.

Secure Communications Over Wireless Broadcast Networks: Stability and Utility Maximization

A wireless broadcast network model with secrecy constraints is investigated, in which a source node broadcasts K confidential message flows to K user nodes, with each message intended to be decoded accurately by one user and to be kept secret from all other users (who are thus considered to be eavesdroppers with regard to all other messages but their own). The source maintains a queue for each message flow if it is not served immediately. The channel from the source to the K users is modeled as a fading broadcast channel, and the channel state information is assumed to be known to the source and the corresponding receivers. Two eavesdropping models are considered. For a collaborative eavesdropping model, in which the eavesdroppers exchange their outputs, the secrecy capacity region is obtained, within which each rate vector is achieved by using a time-division scheme and a source power control policy over channel states. A throughput optimal queue-length-based rate scheduling algorithm is further derived that stabilizes all arrival rate vectors contained in the secrecy capacity region. Moreover, the network utility function is maximized via joint design of rate control, rate scheduling, power control, and secure coding. More precisely, a source controls the message arrival rate according to its message queue, the rate scheduling selects a transmission rate based the queue length vector, and the rate vector is achieved by power control and secure coding. These components work jointly to solve the network utility maximization problem. For a noncollaborative eavesdropping model, in which eavesdroppers do not exchange their outputs, an achievable secrecy rate region is derived based on a time-division scheme, and the queue-length-based rate scheduling algorithm and the corresponding power control policy are obtained that stabilize all arrival rate vectors in this region. The network utility maximizing rate control vector is also obtained.

In This Issue

In This Issue

Dynamic pricing and scheduling in a multi-class single-server queueing system

This paper investigates an optimal sequencing and dynamic pricing problem for a two-class queueing system. Using a Markov Decision Process based model, we obtain structural characterizations of optimal policies. In particular, it is shown that the optimal pricing policy depends on the entire queue length vector but some monotonicity results prevail as the composition of this vector changes. A numerical study finds that static pricing policies may have significant suboptimality but simple dynamic pricing policies perform well in most situations.

Averaging Principles for a Diffusion-Scaled, Heavy-Traffic Polling Station with K Job Classes

This paper provides heavy traffic limit theorems for a polling station serving jobs from K exogenous renewal arrival streams. It is a standard result that the limiting diffusion-scaled total workload process is semimartingale reflected Brownian motion. For polling stations, however, no such limit exists in general for the diffusion-scaled, K-dimensional queue length or workload vector processes. Instead, we prove that these processes admit averaging principles, the natures of which depend on the service discipline employed at the polling station. Parameterized families of exhaustive and gated service disciplines are investigated. Each policy under consideration has K stochastic matrices associated with it—one for each job class—that describe the transition of the workload vector while a given job class is being polled. These matrices give rise to K probability vectors that are vertices of a K - 1-simplex. Loosely speaking, each point of this simplex acts as an instantaneous lifting operator, converting the one-dimensional limiting total workload process to the K-dimensional workload vector. The averaging principles—needed because the workload vector traverses a Hamiltonian cycle over K edges of the simplex infinitely fast under diffusion scaling—are limits of cumulative cost functions of the diffusion-scaled workload vector, evaluated over a fixed time interval. The “averaging” takes place over the edges of the simplex.

Discrete time queueing networks with product form steady state. Availability and performance analysis in an integrated model

For a discrete time network of generalized Bernoulli servers with unreliable nodes we derive the steady state probabilities for the joint queue length vector for all nodes and the availability status of the network. This allows us to assess the performance behavior and the reliability, resp. availability, of the network in an integrated model. Because our result exhibits a product form for the steady state distribution it opens the path to fast algorithmic evaluation of the desired performance and reliability indices.

Heavy-traffic analysis of the M/PH/1 discriminatory processor sharing queue with phase-dependent weights

We analyze a generalization of the Discriminatory Processor Sharing (DPS)queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume phase-type distributed service requirements and allow that customers have different weights in various phases of their service. We establish a state-space collapse for the queue length vector in heavy traffic. The result shows that in the limit, the queue length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by [2] who considered a DPS queue with exponentially distributed service requirements. We finally discuss some implications for residual service requirements and monotonicity properties in the ordinary DPS model.

Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing

We analyze a generalization of the discriminatory processor-sharing (DPS) queue in a heavy-traffic setting. Customers present in the system are served simultaneously at rates controlled by a vector of weights. We assume that customers have phase-type distributed service requirements and allow that customers have different weights in various phases of their service. In our main result we establish a state-space collapse for the queue-length vector in heavy traffic. The result shows that in the limit, the queue-length vector is the product of an exponentially distributed random variable and a deterministic vector. This generalizes a previous result by Rege and Sengupta [Rege, K. M., B. Sengupta. 1996. Queue length distribution for the discriminatory processor-sharing queue. Oper. Res. 44(4) 653–657], who considered a DPS queue with exponentially distributed service requirements. Their analysis was based on obtaining all moments of the queue-length distributions by solving systems of linear equations. We undertake a more direct approach by showing that the probability-generating function satisfies a partial differential equation that allows a closed-form solution after passing to the heavy-traffic limit. Making use of the state-space collapse result, we derive interesting properties in heavy traffic: (i) For the DPS queue, we obtain that, conditioned on the number of customers in the system, the residual service requirements are asymptotically independent and distributed according to the forward recurrence times. (ii) We then investigate how the choice for the weights influences the asymptotic performance of the system. In particular, for the DPS queue we show that the scaled holding cost reduces as classes with a higher value for dk/E(Bkfwd) obtain a larger share of the capacity, where dk is the cost associated to class k, and E(Bkfwd) is the forward recurrence time of the class-k service requirement. The applicability of this result for a moderately loaded system is investigated by numerical experiments.

Optimal Transmission Scheduling in Symmetric Communication Models With Intermittent Connectivity

We consider a slotted system with N queues, and independent and identically distributed (i.i.d.) Bernoulli arrivals at each queue during each slot. Each queue is associated with a channel that changes between "on" and "off" states according to i.i.d. Bernoulli processes. We assume that the system has K identical transmitters ("servers"). Each server, during each slot, can transmit up to C packets from each queue associated with an "on" channel. We show that a policy that assigns the servers to the longest queues whose channel is "on" minimizes the total queue size, as well as a broad class of other performance criteria. We provide several extensions, as well as some qualitative results for the limiting case where N is very large. Finally, we consider a "fluid" model under which fractional packets can be served, and subject to a constraint that at most C packets can be served in total from all of the N queues. We show that when K=N, there is an optimal policy which serves the queues so that the resulting vector of queue lengths is "Most Balanced" (MB)

Queueing systems with leadtime constraints: A fluid-model approach for admission and sequencing control

We study how multi-product queueing systems should be controlled so that sojourn times (or end-to-end delays) do not exceed specified leadtimes. The network dynamically decides when to admit new arrivals and how to sequence the jobs in the system. To analyze this difficult problem, we propose an approach based on fluid-model analysis that translates the leadtime specifications into deterministic constraints on the queue length vector. The main benefit of this approach is that it is possible (and relatively easy) to construct scheduling and multi-product admission policies for leadtime control. Additional results are: (a) While this approach is simpler than a heavy-traffic approach, the admission policies that emerge from it are also more specific than, but consistent with, those from heavy-traffic analysis. (b) A simulation study gives a first indication that the policies also perform well in stochastic systems. (c) Our approach specifies a “tailored” admission region for any given sequencing policy. Such joint admission and sequencing control is “robust” in the following sense: system performance is relatively insensitive to the particular choice of sequencing rule when used in conjunction with tailored admission control. As an example, we discuss the tailored admission regions for two well-known sequencing policies: Generalized Processor Sharing and Generalized Longest Queue. (d) While we first focus on the multi-product single server system, we do extend to networks and identify some subtleties.

On the Correlation Structure of Closed Queueing Networks

Closed networks of exponential queues are prototypes of systems which show strong negative dependence properties. We use the device of a test customer traveling in the network to evaluate the qualitative behavior of the network's correlation. The results are: negative association of successive sojourn times during a cycle; negative correlation of two successive cycle times; feedback-sojourn times are positively associated; correlation between cycles vanishes over long distances geometrically fast. Generalizations to networks with general topology are given. Additionally we prove for cyclic networks that the conditional cycle time stochastically increases when the initial joint queue length vector increases with respect to the partial sum ordering. This connects the space–time correlation with space–time monotonicity.