We consider a polling system with two queues, exhaustive service, no switch-over times and exponential service times. The waiting cost depends on the position of the queue relative to the server: It costs a customer c per time unit to wait in the busy queue (where the server is) and d per time unit in the idle queue (where no server is). Customers arrive according to a Poisson process. We study the control problem of how arrivals should be routed to the two queues in order to minimize expected waiting costs and characterize individually and socially optimal routing policies under three scenarios of available information at decision epochs: no, partial and complete information. In the complete information case, we develop a new iterative algorithm to determine individually optimal policies, and show that such policies can be described by a switching curve. We conjecture that a linear switching curve is socially optimal, and prove that this policy is indeed optimal for the fluid version of the two-queue polling system.
The paper analyzes the performance of tandem network of polling queue with setups. For a system with two-products and two-stations, we propose a new approach based on a partially-collapsible state-space characterization to reduce state-space complexity. In this approach, the size of the state-space is varied depending on the information needed to determine buffer levels and waiting times. We evaluate system performance under different system setting and comment on the numerical accuracy of the approach as well as provide managerial insights. Numerical results show that approach yields reliable…
In this paper, we analyze a two-queue random time-limited Markov modulated polling model. In the first part of the paper, we investigate the fluid version: Fluid arrives at the two queues as two independent flows with deterministic rate. There is a single server that serves both queues at constant speeds. The server spends an exponentially distributed amount of time in each queue. After the completion of such a visit time to one queue, the server instantly switches to the other queue, i.e., there is no switchover time. For this model, we first derive the Laplace-Stieltjes Transform (LST) of th…
In 1987, J.W. Cohen analyzed the so-called Serve the Longest Queue (SLQ) queueing system, where a single server attends two non-symmetric $M/G/1$-type queues, exercising a non-preemptive priority switching policy. Cohen further analyzed in 1998 a non-symmetric 2-queue Markovian system, where newly arriving customers follow the Join the Shortest Queue (JSQ) discipline. The current paper generalizes and extends Cohen's works by studying a combined JSQ-SLQ model, and by broadening the scope of analysis to a non-symmetric 3-queue system, where arriving customers follow the JSQ strategy and a singl…
We study an optimal-control problem of polling systems with large switchover times, when a holding cost is incurred on the queues. In particular, we consider a stochastic network with a single server that switches between several buffers (queues) according to a pre-specified order, assuming that the switchover times between the queues are large relative to the processing times of individual jobs. Due to its complexity, computing an optimal control for such a system is prohibitive, and so we instead search for an asymptotically optimal control. To this end, we first solve an optimal control pro…
我们考虑一个一般的轮询模型与$N$站。车站服务详尽，并在循环顺序。一旦一个工作站队列变为空，服务器不会立即切换到下一个工作站。相反，它在站点上等待新工作的可能到达（“等待和观望”），并且在这种情况下，它以穷举的方式重新启动服务。服务器空闲等待的总时间被设置为每个站的固定的确定性参数。允许切换时间和服务时间分别遵循某种一般分布。在某些情况下，其特征在于，该策略产生比穷举策略严格更低的平均排队延迟，穷举策略对应于将所有站的“等待观望信用”设置为等于零。这扩展了Peköz（Probability in the Engineering and Informational Sciences 13（1999））和Boxma et al.（Annals of运筹学112（2002））的结果。此外，我们给出了一个下界的延迟{ \it所有}的策略，允许服务器等待在车站，即使没有工作。
We establish sufficient conditions for the existence of moments of the steady-state queue in polling systems operating under the binomial-exhaustive policy (BEP). We assume that the server switches between the different buffers according to a pre-specified table, and that switchover times are incurred whenever the server moves from one buffer to the next. We further assume that customers arrive according to independent Poisson processes, and that the service and switchover times are independent random variables with general distributions. We then propose a simple scheme to approximate the mome…
Load balancing in parallel infinite-server queues with action delay via phase representation
Graduate School of Science and Technology, University of Tsukuba（筑波大学理工学研究科）；Institute of Systems and Information Engineering, University of Tsukuba（筑波大学系统信息工学研究院）
We develop a robust queueing network analyzer algorithm to approximate the steady-state performance of a single-class open queueing network of single-server queues with Markovian routing. The algorithm allows non-renewal external arrival processes, general service-time distributions and customer feedback. We focus on the customer flows, defined as the continuous-time processes counting customers flowing into or out of the network, or flowing from one queue to another. Each flow is partially characterized by its rate and a continuous function that measures the stochastic variability over time. …
Cryptocurrency networks such as Bitcoin have emerged as a distributed alternative to traditional centralized financial transaction networks. However, there are major challenges in scaling up the throughput of such networks. Lightning network and Spider network are alternates that build bidirectional payment channels on top of cryptocurrency networks using smart contracts, to enable fast transactions that bypass the Blockchain. In this paper, we study the problem of routing transactions in such a payment processing network. We first propose a Stochastic model to study such a system, as opposed …
We consider many-server queueing systems with heterogeneous exponential servers and renewal arrivals. The service rate of each server is a random variable drawn from a given distribution. We develop a framework for analyzing the heavy traffic limit of these queues in random environment using probability measure-valued stochastic processes. We introduce the measure-valued fairness process which denotes the proportion of cumulative idleness experienced by servers whose rates fall in a Borel subset of the support of the service rates. It can be shown that these fairness processes do not converge …
