Time-Limited and k-Limited Polling Systems: A Matrix Analytic Solution

In this paper, we will develop a tool to analyze polling systems with the autonomous-server, the time-limited, and the k-limited service discipline. It is known that these disciplines do not satisfy the well-known branching property in polling system, therefore, hardly any exact result exists in the literature for them. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-length at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains. Finally, we will show that our tool works also in the case of a tandem queueing network with a single server that can serve one queue at a time

Time-limited and k-limited polling systems: a matrix analytic solution

In this paper, we will develop a tool to analyze polling systems with the autonomous-server, the time-limited, and the k-limited service discipline. It is known that these disciplines do not satisfy the well-known branching property in polling system, therefore, hardly any exact result exists in the literature for them. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-length at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains. Finally, we will show that our tool works also in the case of a tandem queueing network with a single server that can serve one queue at a time

The asymptotic variance of departures in critically loaded queues

We consider the asymptotic variance of the departure counting process D(t) of the GI/G/1 queue; D(t) denotes the number of departures up to time t. We focus on the case that the system load rho equals 1, and prove that the asymptotic variance rate satisfies lim_t Var D(t)/t = lambda (1 - 2/pi) (c_a2 + c_s2) where lambda is the arrival rate and c_a2 and c_s2 are squared coefficients of variation of the inter-arrival and service times respectively. As a consequence, the departures variability has a remarkable singularity in case rho equals 1, in line with the BRAVO effect (Balancing Reduces Asymptotic Variance of Outputs) which was previously encountered in the finite-capacity birth-death queues. Under certain technical conditions, our result generalizes to multi-server queues, as well as to queues with more general arrival and service patterns. For the M/M/1 queue we present an explicit expression of the variance of D(t) for any t

A transient analysis of polling systems operating under exponential time-limited service disciplines

In the present article, we analyze a class of time-limited polling systems. In particular, we will derive a direct relation for the evolution of the joint queue-length during the course of a server visit. This will be done both for the pure and the exhaustive exponential time-limited discipline for general service time requirements and preemptive service. More specifically, service of individual customers is according to the preemptive-repeat-random strategy, i.e., if a service is interrupted, then at the next server visit a new service time will be drawn from the original service-time distribution. Moreover, we incorporate customer routing in our analysis, such that it may be applied to a large variety of queueing networks with a single server operating under one of the before-mentioned time-limited service disciplines. We study the time-limited disciplines by performing a transient analysis for the queue length at the served queue. The analysis of the pure time-limited discipline builds on several known results for the transient analysis of the M/G/1 queue. Besides, for the analysis of the exhaustive discipline, we will derive several new results for the transient analysis of an M/G/1 during a busy period. The final expressions (both for the exhaustive and pure case) that we obtain for the key relations generalize previous results by incorporating customer routing or by relaxing the exponentiality assumption on the service times. Finally, based on the interpretation of these key relations, we formulate a conjecture for the key relation for any branching-type service discipline operating under an exponential time-limit

Delay in a tandem queueing model with mobile queues: An analytical approximation

In this paper, we analyze the end-to-end delay performance of a tandem queueing system with mobile queues. Due to state-space explosion there is no hope for a numerical exact analysis for the joint-queue length distribution. For this reason, we present an analytical approximation that is based on queue length analysis. Through extensive numerical validation, we nd that the queue length approximation exhibits excellent performance for light tra c load

Online Self-Organizing Network Control with Time Averaged Weighted Throughput Objective

We study an online multisource multisink queueing network control problem characterized with self-organizing network structure and self-organizing job routing. We decompose the self-organizing queueing network control problem into a series of interrelated Markov Decision Processes and construct a control decision model for them based on the coupled reinforcement learning (RL) architecture. To maximize the mean time averaged weighted throughput of the jobs through the network, we propose a reinforcement learning algorithm with time averaged reward to deal with the control decision model and obtain a control policy integrating the jobs routing selection strategy and the jobs sequencing strategy. Computational experiments verify the learning ability and the effectiveness of the proposed reinforcement learning algorithm applied in the investigated self-organizing network control problem

Time-limited polling systems with batch arrivals and phase-type service times

In this paper, we develop a general framework to analyze polling systems with either the autonomous-server or the time-limited service discipline. According to the autonomous-server discipline, the server continues servicing a queue for a certain period of time. According to the time-limited service discipline, the server continues servicing a queue for a certain period of time or until the queue becomes empty, whichever occurs first. We consider Poisson batch arrivals and phase-type service times. It is known that these disciplines do not satisfy the well-known branching property in polling systems. Therefore, hardly any exact results exist in the literature. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-lengths at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains

Delay in a tandem queueing model with mobile queues : an analytical approximation

Abstract In this paper, we analyze the end-to-end delay performance of a tandem queueing system with mobile queues. Due to state-space explosion there is no hope for a numerical exact analysis for the joint-queue length distribution. For this reason, we present an analytical approximation that is based on queue length analysis. Through extensive numerical validation, we find that the queue length approximation exhibits excellent performance for light and moderate traffic load

A tandem queueing model for delay analysis in disconnected ad hoc networks

Ad hoc network routing protocols may fail to operate in the absence of an end-to-end connection from source to destination. This deficiency can be resolved by so-called delay-tolerant networking which exploits the mobility of the nodes by letting them operate as relays according to the store-carry-and-forward paradigm. In this work, we analyze the delay performance of a small mobile ad hoc network by considering a tandem queueing system. We present an exact packet-level analysis by applying ideas from the polling literature. Due to the state-space expansion, this analysis cannot efficiently be applied for all model parameter settings. For this reason, an analytical approximation is constructed and its excellent performance has extensively been validated. Numerical results on the mean end-to-end delay show that the switch-over time distribution impacts this metric only through its first two moments. Finally, we study delay optimization under power control