A polling model with threshold switching

Queue lengths and workloads in polling systems

In this paper we consider a ring of N ≥ 1 queues served by a single server in a cyclic order. After having served a queue (according to a service discipline that may vary from queue to queue), there is a switch-over period and then the server serves the next queue and so forth. This model is known in the literature as a polling model. Each of the queues is fed by a non-decreasing Lévy process, which can be different during each of the consecutive periods within the server’s cycle. The N-dimensional Lévy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams. For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch. Our analysis heavily relies on establishing a link between fluid (Lévy input) polling systems and multi-type Jiřina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the branching property for a discipline, which can be traced back to Fuhrmann and Resing. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated.

In this paper we consider a ring of $N\\ge 1$ queues served by a single server\nin a cyclic order. After having served a queue (according to a service\ndiscipline that may vary from queue to queue), there is a switch-over period\nand then the server serves the next queue and so forth. This model is known in\nthe literature as a \\textit{polling model}. Each of the queues is fed by a\nnon-decreasing L\\'evy process, which can be different during each of the\nconsecutive periods within the server's cycle. The $N$-dimensional L\\'evy\nprocesses obtained in this fashion are described by their (joint) Laplace\nexponent, thus allowing for non-independent input streams. For such a system we\nderive the steady-state distribution of the joint workload at embedded epochs,\ni.e. polling and switching instants. Using the Kella-Whitt martingale, we also\nderive the steady-state distribution at an arbitrary epoch. Our analysis\nheavily relies on establishing a link between fluid (L\\'evy input) polling\nsystems and multi-type Ji\\v{r}ina processes (continuous-state discrete-time\nbranching processes). This is done by properly defining the notion of the\n\\textit{branching property} for a discipline, which can be traced back to\nFuhrmann and Resing. This definition is broad enough to contain the most\nimportant service disciplines, like exhaustive and gated.\n

We consider a class of two-queue polling systems with exhaustive service, where the order in which the server visits the queues is governed by a discrete-time Markov chain. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we obtain explicit expressions for the Laplace---Stieltjes transforms of the waiting-time distributions and the probability generating function of the joint queue length distribution at an arbitrary point in time. We also study the heavy-traffic behaviour of properly scaled versions of these distributions, which results in compact and closed-form expressions for the distribution functions themselves. The heavy-traffic behaviour turns out to be similar to that of cyclic polling models, provides insights into the main effects of the model parameters when the system is heavily loaded, and can be used to derive closed-form approximations for the waiting-time distribution or the queue length distribution.

This thesis presents models for the performance analysis of a recent communication paradigm: mobile ad hoc networking. The objective of mobile ad hoc networking is to provide wireless connectivity between stations in a highly dynamic environment. These dynamics are driven by the mobility of stations and by breakdowns of stations, and may lead to temporary disconnectivity of parts of the network. Applications of this novel paradigm can be found in telecommunication services, but also in manufacturing systems, road-traffic control, animal monitoring and emergency networking. The performance of mobile ad hoc networks in terms of buffer occupancy and delay is quantified in this thesis by employing specific queueing models, viz., time-limited polling models. These polling models capture the uncontrollable characteristic of link availability in mobile ad hoc networks. Particularly, a novel, so-called pure exponential time-limited, service discipline is introduced in the context of polling systems. The highlighted performance characteristics for these polling systems include the stability, the queue lengths and the sojourn times of the customers. Stability conditions prescribe limits on the amount of tra±c that can be sustained by the system, so that the establishment of these conditions is a fundamental keystone in the analysis of polling models. Moreover, both exact and approximate analysis is presented for the queue length and sojourn time in time-limited polling systems with a single server. These exact analytical techniques are extended to multi-server polling systems operating under the pure time-limited service discipline. Such polling systems with multiple servers effectively may reflect large communication networks with multiple simultaneously active links, while the systems with a single server represent performance models for small networks in which a single communication link can be active at a time.

Markovian polling systems with an application to wireless random-access networks

This study is devoted to a queueing analysis of polling systems with a probabilistic server routing mechanism. A single server serves a number of queues, switching between the queues according to a discrete time parameter Markov chain. The switchover times between queues are nonneghgible. It is observed that the total amount of work in this Markovian polling system can be decomposed into two independent parts, viz., (i) the total amount of work in the corresponding system without switchover times and (ii) the amount of work in the system at some epoch covered by a switching interval. This work decomposition leads to a pseudoconservation law for mean waiting times, i.e., an exact expression for a weighted sum of the mean waiting times at all queues. The results generalize known results for polling systems with strictly cyclic service.

In this paper we derive decomposition results for the number of customers in polling systems under arbitrary (dynamic) polling order and service policies. Furthermore, we obtain sharper decomposition results for both the number of customers in the system and the waiting times under static polling policies. Our analysis, which is based on distributional laws, relaxes the Poisson assumption that characterizes the polling systems literature. In particular, we obtain exact decomposition results for systems with either Mixed Generalized Erlang (MGE) arrival processes, or asymptotically exact decomposition results for systems with general renewal arrival processes under heavy traffic conditions. The derived decomposition results can be used to obtain the performance analysis of specific systems. As an example, we evaluate the performance of gated Markovian polling systems operating under heavy traffic conditions. We also provide numerical evidence that our heavy traffic analysis is very accurate even for moderate traffic.

In this paper we consider a single-server, cyclic polling system with switch-over times. A distinguishing feature of the model is that the rates of the Poisson arrival processes at the various queues depend on the server location. For this model we study the joint queue length distribution at polling epochs and at the server’s departure epochs. We also study the marginal queue length distribution at arrival epochs, as well as at arbitrary epochs (which is not the same in general, since we cannot use the PASTA property). A generalised version of the distributional form of Little’s law is applied to the joint queue length distribution at customer’s departure epochs in order to find the waiting time distribution for each customer type. We also provide an alternative, more efficient way to determine the mean queue lengths and mean waiting times, using Mean Value Analysis. Furthermore, we show that under certain conditions a Pseudo-Conservation Law for the total amount of work in the system holds. Finally, typical features of the model under consideration are demonstrated in several numerical examples.

We consider multiclass Markovian polling systems with feedback and analyze their average performance measures. Scheduling in polling systems has many applications in computer and communication systems. We utilize the framework that has been effectively used to analyze various composite scheduling algorithms in many types of multiclass queues systematically in conjunction with the functional computation method (Hirayama in Naval Research Logistics 50:719–741, 2003; Journal of the Operations Research Society of Japan 48:226–255, 2005; Advances in queueing theory and network applications, pp. 119–146, Springer, New York, 2009a; Journal of Industrial and Management Optimization 6:541–568, 2010). We define the conditional expected values of the performance measures such as the sojourn times as functions of the system state and find their expressions by solving some equations. Then from these expressions, we derive the average numbers of customers and the average sojourn times for all service stages of customers circulating the system. We consider their application to a packet scheduling problem where multiple categories of packets share a resource.

We consider a model of two M/M/1 queues, served by a single server. The service policy for this polling model is of threshold type: the server serves queue 1 exhaustively, and does not remain at an empty queue if the other one is non-empty. It switches from queue 2 to queue 1 when the size of the latter queue reaches some level T, either preemptively or non-preemptively. All switches are instantaneous.

Dynamic server assignment in a two-queue model

We consider polling models consisting of a single server that visits the queues in a cyclic order. In the vast majority of papers that have appeared on polling models, it is assumed that at each of the individual queues, the customers are served on a first-come-first-served (FCFS) basis. In this paper, we study polling models where the local scheduling policy is not FCFS but instead is varied as last-come-first-served (LCFS), random order of service (ROS), processor sharing (PS), and shortest-job-first (SJF). The service policies are assumed to be either gated or globally gated. The main result of the paper is the derivation of asymptotic closed-form expressions for the Laplace---Stieltjes transform of the scaled waiting-time and sojourn-time distributions under heavy-traffic assumptions. For FCFS service, the asymptotic sojourn-time distribution is known to be of the form $$U \varGamma $$UΓ, where $$U$$U and $$\varGamma $$Γ are uniformly and gamma distributed with known parameters. In this paper, we show that the asymptotic sojourn-time distribution (1) for LCFS is also of the form $$U \varGamma $$UΓ, (2) for ROS is of the form $$\tilde{U} \varGamma $$U~Γ, where $$\tilde{U}$$U~ has a trapezoidal distribution, and (3) for PS and SJF is of the form $$\tilde{U}^* \varGamma $$U~?Γ, where $$\tilde{U}^*$$U~? has a generalized trapezoidal distribution. These results are rather intriguing and lead to new fundamental insight into the impact of the local scheduling policy on the performance of polling models. As a by-product, the heavy-traffic results suggest simple closed-form approximations for the complete waiting-time and sojourn-time distributions for stable systems with arbitrary load values. The accuracy of the approximations is evaluated by simulations.

We consider polling systems with multiple coupled servers. We explore the class of systems that allow an exact analysis. For these systems we present distributional results for the waiting time, the marginal queue length, and the joint queue length at polling epochs. The class in question includes several single-queue systems with a varying number of servers, two-queue two-server systems with exhaustive service and exponential service times, as well as infinite-server systems with an arbitrary number of queues, exhaustive or gated service, and deterministic service times.

We present a heavy traffic analysis of a single-server polling model, with the special features of retrials and glue periods. The combination of these features in a polling model typically occurs in certain optical networking models, and in models where customers have a reservation period just before their service period. Just before the server arrives at a station there is some deterministic glue period. Customers (both new arrivals and retrials) arriving at the station during this glue period will be served during the visit of the server. Customers arriving in any other period leave immediately and will retry after an exponentially distributed time. As this model defies a closed-form expression for the queue length distributions, our main focus is on their heavy-traffic asymptotics, both at embedded time points (beginnings of glue periods, visit periods, and switch periods) and at arbitrary time points. We obtain closed-form expressions for the limiting scaled joint queue length distribution in heavy traffic. We show that these results can be used to accurately approximate the performance of the system for the complete spectrum of load values by use of interpolation approximations.