Mixed Polling with Rerouting and Applications

Queueing systems with a single server in which customers wait to be served at a finite number of distinct locations (buffers/queues) are called discrete polling systems. Polling systems in which arrivals of users occur anywhere in a continuum are called continuous polling systems. Often one encounters a combination of the two systems: the users can either arrive in a continuum or wait in a finite set (i.e. wait at a finite number of queues). We call these systems mixed polling systems. Also, in some applications, customers are rerouted to a new location (for another service) after their service is completed. In this work, we study mixed polling systems with rerouting. We obtain their steady state performance by discretization using the known pseudo conservation laws of discrete polling systems. Their stationary expected workload is obtained as a limit of the stationary expected workload of a discrete system. The main tools for our analysis are: a) the fixed point analysis of infinite dimensional operators and; b) the convergence of Riemann sums to an integral. We analyze two applications using our results on mixed polling systems and discuss the optimal system design. We consider a local area network, in which a moving ferry facilitates communication (data transfer) using a wireless link. We also consider a distributed waste collection system and derive the optimal collection point. In both examples, the service requests can arrive anywhere in a subset of the two dimensional plane. Namely, some users arrive in a continuous set while others wait for their service in a finite set. The only polling systems that can model these applications are mixed systems with rerouting as introduced in this manuscript.Comment: to appear in Performance Evaluatio

Waiting times in queueing networks with a single shared server

We study a queueing network with a single shared server that serves the queues in a cyclic order. External customers arrive at the queues according to independent Poisson processes. After completing service, a customer either leaves the system or is routed to another queue. This model is very generic and finds many applications in computer systems, communication networks, manufacturing systems, and robotics. Special cases of the introduced network include well-known polling models, tandem queues, systems with a waiting room, multi-stage models with parallel queues, and many others. A complicating factor of this model is that the internally rerouted customers do not arrive at the various queues according to a Poisson process, causing standard techniques to find waiting-time distributions to fail. In this paper we develop a new method to obtain exact expressions for the Laplace-Stieltjes transforms of the steady-state waiting-time distributions. This method can be applied to a wide variety of models which lacked an analysis of the waiting-time distribution until now

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

On the Stability of a Polling System with an Adaptive Service Mechanism

We consider a single-server cyclic polling system with three queues where the server follows an adaptive rule: if it finds one of queues empty in a given cycle, it decides not to visit that queue in the next cycle. In the case of limited service policies, we prove stability and instability results under some conditions which are sufficient but not necessary, in general. Then we discuss open problems with identifying the exact stability region for models with limited service disciplines: we conjecture that a necessary and sufficient condition for the stability may depend on the whole distributions of the primitive sequences, and illustrate that by examples. We conclude the paper with a section on the stability analysis of a polling system with either gated or exhaustive service disciplines.Comment: 16 page

Random Fluid Limit of an Overloaded Polling Model

In the present paper, we study the evolution of an overloaded cyclic polling model that starts empty. Exploiting a connection with multitype branching processes, we derive fluid asymptotics for the joint queue length process. Under passage to the fluid dynamics, the server switches between the queues infinitely many times in any finite time interval causing frequent oscillatory behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid limit is random. Additionally, we suggest a method that establishes finiteness of moments of the busy period in an M/G/1 queue.Comment: 36 pages, 2 picture

Stability of the Greedy Algorithm on the Circle

We consider a single-server system with service stations in each point of the circle. Customers arrive after exponential times at uniformly-distributed locations. The server moves at finite speed and adopts a greedy routing mechanism. It was conjectured by Coffman and Gilbert in~1987 that the service rate exceeding the arrival rate is a sufficient condition for the system to be positive recurrent, for any value of the speed. In this paper we show that the conjecture holds true