Higher order approximations for tandem queueing networks

In this paper a higher order approximation for single server queues and tandem queueing networks is proposed and studied. Different from the most popular two-moment based approximations in the literature, the higher order approximation uses the higher moments of the interarrival and service distributions in evaluating the performance measures for queueing networks. It is built upon the MacLaurin series analysis, a method that is recently developed to analyze single-node queues, along with the idea of decomposition using higher orders of the moments matched to a distribution. The approximation is computationally flexible in that it can use as many moments of the interarrival and service distributions as desired and produce the corresponding moments for the waiting and interdeparture times. Therefore it can also be used to study several interesting issues that arise in the study of queueing network approximations, such as the effects of higher moments and correlations. Numerical results for single server queues and tandem queueing networks show that this approximation is better than the two-moment based approximations in most cases.     

Stability of multi-class queueing networks with infinite virtual queues

We generalize the standard multi-class queueing network model by allowing both standard queues and infinite virtual queues which have an infinite supply of work. We pose the general problem of finding policies which allow some of the nodes of the network to work with full utilization, and yet keep all the standard queues in the system stable. Toward this end we show that re-entrant lines, systems of two re-entrant lines through two service stations, and rings of service stations can be stabilized with priority policies under certain parameter restrictions. The analysis throughout the paper depends on model and policy and illustrates the difficulty in solving the general problem.     

On stationary tandem queueing networks with job feedback

The class of tandem queueing networks with job feedback is studied under stationarity conditions on the arrival and service times sequences. Each job, after completing service in the last queue, is fed back (rerouted) to the first one, a random number of times, before leaving the system. The average execution time per job is exactly computed, as the number of jobs becomes large, and is minimized under mild conditions. The degree of parallelism achieved in the processing is also computed. The issue of rate-stability of the system is then considered. The network is defined to be rate-stable iff the job departure rate is equal to the job arrival rate; that depends heavily on the dynamic feedback policy we employ to place rerouted jobs in specific places of the front queue buffer of the network. The condition under which the network is rate-stable is specified, and a dynamic feedback policy is constructed, which rate-stabilizes the system under the maximum possible job arrival rate; thus, it maximizes the dynamic throughput of the network. Other related results concerning the performance of tandem networks with feedback are obtained.Research supported in part by grants NSF-DDM-RIA-9010778, NSF-NCR-9116268, NSF-NCR-NYI-9258 507, by an AT&T Foundation grant and a GTE Fellowship.     

Pooling in tandem queueing networks with non-collaborative servers

This paper considers pooling several adjacent stations in a tandem network of single-server stations with finite buffers. When stations are pooled, we assume that the tasks at those stations are pooled but the servers are not. More specifically, each server at the pooled station picks a job from the incoming buffer of the pooled station and conducts all tasks required for that job at the pooled station before that job is placed in the outgoing buffer. For such a system, we provide sufficient conditions on the buffer capacities and service times under which pooling increases the system throughput by means of sample-path comparisons. Our numerical results suggest that pooling in a tandem line generally improves the system throughput—substantially in many cases. Finally, our analytical and numerical results suggest that pooling servers in addition to tasks results in even larger throughput when service rates are additive and the two systems have the same total number of storage spaces.     

Asymptotic analysis for closed multichain queueing networks with bottlenecks

We consider a two-chain exponential queueing network with a large number of customers that consists of one infinite-server (IS) station and two processor-sharing (PS) or FCFS single-server stations. The asymptotic behavior of the partition function is studied for such a network when one or both PS (FCFS) nodes are heavily loaded. The results are derived using methods of multidimensional complex analysis (the theory of homologies and residues) and the saddle-point method.     

Asymptotic periodicity of recurrent flows in infinite networks

We consider a transport process on an infinite network and, using the corresponding flow semigroup as in Dorn (Semigroup Forum 76:341–356, 2008), investigate its long term behavior. Combining methods from functional analysis, graph theory and stochastics, we are able to characterize the networks for which the flow semigroup converges strongly to a periodic group.     

Asymptotic analysis for closed multiclass queueing networks in critical usage

We consider a class of closed multiclass queueing networks containing First-Come-First-Serve (FCFS) and Infinite Server (IS) stations. These networks have a productform solution for their equilibrium probabilities. We study these networks in an asymptotic regime for which the number of customers and the service rates at the FCFS stations go to infinity with the same order. We assume that the regime is in critical usage, whereby the utilizations of the FCFS servers slowly approach one. The asymptotic distribution of the normalized queue lengths is shown to be in many cases a truncated multivariate normal distribution. Traffic conditions for which the normalized queue lengths arealmost asymptotically independent are determined. Asymptotic expansions of utilizations and expected queue lengths are presented. We show through an example how to obtain asymptotic expansions of performance measures when the networks are in mixed usage and how to apply the results to networks with finite data.Supported partially by NSF grant NCR93-04601.     

A decomposition-related throughput property of tandem queueing networks with blocking

A general throughput property of tandem queueing networks with blocking that relates existing decomposition methods to throughput bounds is discussed using the sample path approach.     

Asymptotic behaviour of the tandem queueing system with identical service times at both queues

Consider a tandem queue consisting of two single-server queues in series, with a Poisson arrival process at the first queue and arbitrarily distributed service times, which for any customer are identical in both queues. For this tandem queue, we relate the tail behaviour of the sojourn time distribution and the workload distribution at the second queue to that of the (residual) service time distribution. As a by-result, we prove that both the sojourn time distribution and the workload distribution at the second queue are regularly varying at infinity of index 1−ν, if the service time distribution is regularly varying at infinity of index −ν (ν>1). Furthermore, in the latter case we derive a heavy-traffic limit theorem for the sojourn time S ⁽²⁾ at the second queue when the traffic load ρ↑ 1. It states that, for a particular contraction factor Δ (ρ), the contracted sojourn time Δ (ρ) S ⁽²⁾ converges in distribution to the limit distribution H(·) as ρ↑ 1 where .     

Asymptotic relations in queueing theory

For the GI?G?1 queueing system a number of asymptotic results are reviewed. Discussed are asymptotics related to the time parameter for t → ∞ relaxation times, heavy traffic theory, restricted accessibility with large bounds, approximation by diffusion processes, exponential and regular variation of the tail of the waiting time distribution, limit theorems and extreme value theorems.     

Symmetry property of the throughput in closed tandem queueing networks with finite buffers

In this paper we consider closed tandem queueing networks with finite buffers and blocking before service. With this type of blocking, a server is allowed to start processing a job only if there is an empty space in the next buffer. It was recently conjectured that the throughput of such networks is symmetrical with respect to the population of the network. That is, the throughput of the network with population N is the same as that with population C − N, where C is the total number of buffer spaces in the network. The main purpose of this paper is to prove this result in the case where the service time distributions are of phase type (PH-distribution). The proof is based on the comparison of the sample paths of the network with populations N and C − N. Finally, we also show that this symmetry property is related to a reversibility property of this class of networks.     

Asymptotic connectivity of infinite graphs

We introduce the notion of the asymptotic connectivity of a graph by generalizing to infinite graphs average connectivity as defined by Beineke, Oellermann, and Pippert. Combinatorial and geometric properties of asymptotic connectivity are then explored. In particular, we compute the asymptotic connectivity of a number of planar graphs in order to determine the extent to which this measure correlates with the large-scale geometry of the graph.     

A tandem queueing model with coupled processors

We consider a tandem queue with coupled processors and analyze the two-dimensional Markov process representing the numbers of jobs in the two stations. A functional equation for the generating function of the stationary distribution of this two-dimensional process is derived and solved through the theory of Riemann-Hilbert boundary value problems.     

Ergodicity of Jackson-type queueing networks

This paper gives a pathwise construction of Jackson-type queueing networks allowing the derivation of stability and convergence theorems under general probabilistic assumptions on the driving sequences; namely, it is only assumed that the input process, the service sequences and the routing mechanism are jointly stationary and ergodic in a sense that is made precise in the paper. The main tools for these results are the subadditive ergodic theorem, which is used to derive a strong law of large numbers, and basic theorems on monotone stochastic recursive sequences. The techniques which are proposed here apply to other and more general classes of discrete event systems, like Petri nets or GSMPs. The paper also provides new results on the Jackson-type networks with i.i.d. driving sequences which were studied in the past.The work of this author was supported in part by a grant from the European Commission DG XIII, under the BRA Qmips contract.The work of this author was supported by a sabbatical grant from INRIA Sophia Antipolis.     

Instability of LIFO queueing networks

Queueing Systems - Under the last-in, first-out (LIFO) discipline, jobs arriving later at a class always receive priority of service over earlier arrivals at any class belonging to the same...     

Monotone control of queueing networks

This paper uses submodularity to obtain monotonicity results for a class of Markovian queueing network service rate control problems. Nonlinear costs of queueing and service are allowed. In contrast to Weber and Stidham [14], our monotonicity theorem considers arbitrary directions in the state space (not just control directions), arrival routing problems, and certain uncontrolled service rates. We also show that, without service costs, transition-monotone controls can be described by simple control regions and switching functions. The theory is applied to queueing networks that arise in a manufacturing system that produces to a forecast of customer demand, and also to assembly and disassembly networks.     

Consistency results on infinite graphs

Consistently there exist ℵ₂-chromatic graphs with no ℵ₁-chromatic subgraphs. The statement that every uncountably chromatic graph of size ℵ₁ contains an uncountably chromaticω-connected subgraph is consistent and independent. It is consistent that there is an uncountably chromatic graph of size ℵ_{ω 1} in which every subgraph with size less than ℵ_{ω 1} is countably chromatic. Partially supported by Hungarian Science Research Fund Nu. 1805.