Approximation Analysis of Open Acyclic Exponential Queueing Networks with Blocking

In the last chapter, we used Markov process models for queueing systems with Poisson arrivals and exponential service times. To model a system as a Markov process, we should be able to give complete distribution characteristics of the process beyond time t, using what we know about the process at t and changes that may occur after t, without referring back to the events before t. When arrivals are Poisson and service times are exponential, because of the memoryless property of the exponential distribution we are able to use the Markov process as a model. If the arrival rate is λ and service rate is μ, at any time point t, time to next arrival has the exponential distribution with rate λ, and if a service is in progress, the remaining service time has the exponential distribution with rate μ. If one or both of the arrival and service distributions are non-exponential, the memoryless property does not hold and a Markov model of the type discussed in the last chapter does not work. In this chapter, we discuss a method by which a Markov model can be constructed, not for all t, but for specific time points on the time axis.KeywordsMarkov ChainService TimeTransition Probability MatrixProbability Generate FunctionService Time DistributionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the dynamic assignment of cross‐trained servers to stations in understaffed lines with finite buffers. Our objective is to maximize the production rate. We identify optimal server assignment policies for systems with three stations, two servers, different flexibility structures, and either deterministic service times and arbitrary buffers or exponential service times and small buffers. We use these policies to develop server assignment heuristics for Markovian systems with larger buffer sizes that appear to yield near‐optimal throughput. In the deterministic setting, we prove that the best possible production rate with full server flexibility and infinite buffers can be attained with partial flexibility and zero buffers, and we identify the critical skills required to achieve this goal. We then present numerical results showing that these critical skills, employed with an effective server assignment policy, also yield near‐optimal throughput in the Markovian setting, even for small buffer sizes. Thus, our results suggest that partial flexibility is sufficient for near‐optimal performance, and that flexibility structures that are effective for deterministic and infinite‐buffered systems are also likely to perform well for finite‐buffered stochastic systems.

In Chapter 3 we developed regenerative simulation methods for networks of queues with exponential service times and Markovian job routing. In this chapter, we derive analogous estimation procedures for networks with general service times. The development requires that we restrict attention to networks that have a “single state” in which all jobs are at the same service center with exactly one job in service. Regenerative cycles are defined in terms of entrances to the single state.

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all work conserving policies.However, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations we prove that if the fluid model is not weakly stable under the class of all work conserving policies, then any corresponding queueing network is not rate stable under the class of all work conserving policies. We establish the result by building a particular work conserving scheduling policy which makes any corresponding stochastic process transient. An important corollary of our result is that the condition of the form ρ* < 1, which was proven in [7) to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for any corresponding queueing network. Here ρ* is a certain computable parameter of the network involving virtual station and push start conditions.

The fluid model has proven to be one of the most effective tools for the analysis of stochastic queueing networks, specifically for the analysis of stability. It is known that stability of a fluid model implies positive (Harris) recurrence (stability) of a corresponding stochastic queueing network, and weak stability implies rate stability of a corresponding stochastic network. These results have been established both for cases of specific scheduling policies and for the class of all nonidling policies.\n¶\nHowever, only partial converse results have been established and in certain cases converse statements do not hold. In this paper we close one of the existing gaps. For the case of networks with two stations, we prove that if the fluid model is not weakly stable under the class of all nonidling policies, then a corresponding queueing network is not rate stable under the class of all nonidling policies. We establish the result by building a particular nonidling scheduling policy which makes the associated stochastic process transient. An important corollary of our result is that the condition ρ<sup>*</sup>≤1, which was proven in [Oper. Res. 48 (2000) 721–744] to be the exact condition for global weak stability of the fluid model, is also the exact global rate stability condition for an associated queueing network. Here ρ<sup>*</sup> is a certain computable parameter of the network involving virtual station and push start conditions.

Estimating the performance measures of queues and queueing networks with finite buffers and general service times is a complex and challenging problem. This paper presents an effective two-moment approach for calculating the state probability distribution of the performance measures of the queues in these systems. Performance measures of throughput (), the work-in-process (WIP) and sojourn or cycle time () are estimated. The generalised expansion method is employed to calculate the network performance measures of the topologies and extensive computational experiments are provided. It is shown that the performance measures of WIP and sojourn time for multi-server exponential systems have an s-shaped function and this measure is extended to networks. It is argued that this relates directly to the system capacity of queues in finite buffer queueing networks. Series, merge and splitting topologies of finite buffer multi-server queueing networks are then examined. Numerous experiments involving the given topological configurations are demonstrated and the s-shaped behaviour of the overall system performance measures for system capacity are explained.

We show how to calculate the Age of Information in an overtake-free network of quasi-reversible queues, with exponential exogenous interarrivals of multiple classes of update packets and exponential service times at all nodes. Results are provided for any number of M/M/1 First-Come-First-Served (FCFS) queues in tandem, and for a network with two classes of update packets, entering through different queues in the network and exiting through the same queue. The main takeaway is that in a network with different classes of update packets, individual classes roughly preserve the ages they would achieve if they were alone in the network, except when shared queues become saturated, in which case the ages increase considerably.

We address the approximate analysis of open networks of PH¦PH¦1 and PH¦PH¦1¦K queues. We start from the analysis of open queueing networks (QNs) as proposed by Whitt, where large QNs are decomposed into individual GI¦G¦1 queues, characterized by the first and second moment of the service and interarrival time distribution. We extend this approach in two ways.First of all, we use PH¦PH¦1 queues, instead of GI¦G¦1 queues, so that the individual queues can be solved exactly, using matrix-geometric techniques. Secondly, we allow for the inclusion of finite-buffer queues. In doing so, the proposed decomposition becomes an iterative process.We present the mathematical background of the approach as well as a tool implementation (Qnaut). It turns out that our approach not only yields accurate results (within a few percents from simulation results) but also is very fast in obtaining them (in comparison with simulation).

In this paper, we analyze a two station tandem queue with Poisson arrivals and exponential service times. All arrivals occur at the first line, and, after receiving service at the first station, they proceed to the second line. There is only a finite buffer between the stations, and, as soon as the buffer is full, any job completed by the first server is lost. To reduce customer loss, the first server can move to the second station and help the second server, thereby increasing its service rate. Once the work at station two is complete, the job leaves the system. The problem will be solved by using eigenvalues which can be obtained in explicit form. It is shown that this method is substantially faster than matrix analytic methods.

Paid curbside parking can be advantageously modeled as a network of interdependent queues. To this end, we introduce methods for analyzing a special class of networks of finite capacity queues where drivers arrive from an exogenous source, join the queue if there is an available parking space, or continue to search at an adjacent queue for an available space. Furthermore, we apply this model to estimate the proportion of drivers cruising in the neighborhood of Belltown, Seattle, WA, USA. Using occupancy approximated by parking transaction data, we estimate the percentage of drivers cruising for curbside parking by comparing the rate of drivers unable to find parking to bulk through-traffic measurement data. We find percentages of up to 50% for a Belltown's 1st Ave. depending on the time, day, and direction of travel. We then calculate a per vehicle travel-time cost to social welfare incurred by this proportion: upward of a 10% increase in travel time to all drivers along 1st Ave. Last, we introduce a simulation tool and test assumptions made when estimating interesting performance metrics like the probability of a block-face being full. Our results suggest that while assuming exponential service time distributions is not justified, mean rate solutions under a Markovian relaxation of the problem is comparable to service times representative of parking transaction data in simulation.

Customers arriving randomly are served by a queueing system consisting of a sequence of two service stations with infinite queue allowable before the first station and no queue allowable between the stations. The moment generating functions of the steady-state queueing times as well as the generating functions of the steady-state numbers of customers in the various parts of the system are obtained under assumptions of Poisson process of arrivals and arbitrarily distributed service times at both stations. The cases of regular and exponential service times are investigated in some more detail, and the results obtained are extended to include the queueing system with a sequence of two stations with a finite intermediate queue allowable between them, infinite queue allowed before the first station, Poisson process of arrivals and regular service times at both stations.

Models are given for sequences of correlated exponential interarrival and service times for a single-server queue. These multivariate exponential models are formed as probabilistic linear combinations of sequences of independent exponential random variables and are easy to generate on a computer. Limiting results for customer waiting time under heavy traffic conditions are obtained for these queues. Heavy traffic results are useful for analyzing the effect of correlated interarrival and service times in queues on such quantities as queue length and customer waiting time. They can also be used to check simulation results.

This paper investigates theoretical properties of throughput and cost in linear loss networks. The maximum throughput of the network with exponential service times is derived and the arrival process that maximizes throughput, given a fixed arrival rate, is established. For general service times, an asymptotically critical loading regime is identified such that the probability of an arbitrary customer being lost is strictly within (0,1) as the network size increases. This regime delivers throughput comparable to the maximum at a relatively low network cost. The paper establishes the asymptotic throughput and network cost under this critical loading.

This paper is concerned with the Bayesian analysis of general queues with Poisson input and exponential service times. Joint posterior distribution of the arrival rate and the individual service rate is obtained from a sample consisting inn observations of the interarrival process andm complete service times. Posterior distribution of traffic intensity inM/M/c is also obtained and the statistical analysis of the ergodic condition from a decision point of view is discussed.

short-paper Share on Tail asymptotics of the stationary distribution for a two-node generalized Jackson network Author: Masakiyo Miyazawa Tokyo University of Science, Noda, Chiba, Japan Tokyo University of Science, Noda, Chiba, JapanView Profile Authors Info & Claims ACM SIGMETRICS Performance Evaluation ReviewVolume 42Issue 2September 2014pp 70–72https://doi.org/10.1145/2667522.2667545Published:04 September 2014Publication History 1citation26DownloadsMetricsTotal Citations1Total Downloads26Last 12 Months0Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteGet Access