Stochastic bounds for a polling system

In this note we consider two queueing systems: a symmetric polling system with gated service at allN queues and with switchover times, and a single-server single-queue model with one arrival stream of ordinary customers andN additional permanently present customers. It is assumed that the combined arrival process at the queues of the polling system coincides with the arrival process of the ordinary customers in the single-queue model, and that the service time and switchover time distributions of the polling model coincide with the service time distributions of the ordinary and permanent customers, respectively, in the single-queue model. A complete equivalence between both models is accomplished by the following queue insertion of arriving customers. In the single-queue model, an arriving ordinary customer occupies with probabilityp _i a position at the end of the queue section behind theith permanent customer,i = l, ...,N. In the cyclic polling model, an arriving customer with probabilityp _i joins the end of theith queue to be visited by the server, measured from its present position.For the single-queue model we prove that, if two queue insertion distributions {p _{i, i} = l, ...,N} and {q _{i, i} = l, ...,N} are stochastically ordered, then also the workload and queue length distributions in the corresponding two single-queue versions are stochastically ordered. This immediately leads to equivalent stochastic orderings in polling models.Finally, the single-queue model with Poisson arrivals andp ₁ = 1 is studied in detail.Part of the research of the first author has been supported by the Esprit BRA project QMIPS.     

Stability,monotonicity and invariant quantities in general polling systems

Consider a polling system withK1 queues and a single server that visits the queues in a cyclic order. The polling discipline in each queue is of general gated-type or exhaustive-type. We assume that in each queue the arrival times form a Poisson process, and that the service times, the walking times, as well as the set-up times form sequences of independent and identically distributed random variables. For such a system, we provide a sufficient condition under which the vector of queue lengths is stable. We treat several criteria for stability: the ergodicity of the process, the geometric ergodicity, and the geometric rate of convergence of the first moment. The ergodicity implies the weak convergence of station times, intervisit times and cycle times. Next, we show that the queue lengths, station times, intervisit times and cycle times are stochastically increasing in arrival rates, in service times, in walking times and in setup times. The stability conditions and the stochastic monotonicity results are extended to the polling systems with additional customer routing between the queues, as well as bulk and correlated arrivals. Finally, we prove that the mean cycle time, the mean intervisit time and the mean station times are invariant under general service disciplines and general stationary arrival and service processes.     

The 1-Center and 1-Highway problem revisited

We consider a two-queue polling model in which customers upon arrival join the shorter of two queues. Customers arrive according to a Poisson process and the service times in both queues are independent and identically distributed random variables having the exponential distribution. The two-dimensional process of the numbers of customers at the queue where the server is and at the other queue is a two-dimensional Markov process. We derive its equilibrium distribution using two methodologies: the compensation approach and a reduction to a boundary value problem.     

Dynamic server assignment in a two-queue model

We consider a polling model of two M/G/1 queues, served by a single server. The service policy for this polling model is of threshold type. Service at queue 1 is exhaustive. Service at queue 2 is exhaustive unless the size of queue 1 reaches some level T during a service at queue 2; in the latter case the server switches to queue 1 at the end of that service. Both zero- and nonzero switchover times are considered. We derive exact expressions for the joint queue length distribution at customer departure epochs, and for the steady-state queue-length and sojourn time distributions. In addition, we supply a simple and very accurate approximation for the mean queue lengths, which is suitable for optimization purposes.     

Polling systems with state-dependent setup times

We analyze a polling system with multiple stations (queues) attended by a cycling server, in which a setup occurs only when the queue that is polled by the server has one or more customers present. Although such systems are appropriate for modeling numerous manufacturing and telecommunication systems, their analysis is not well developed in the literature. We provide an exact analysis for the 2 station model and present two approximation schemes to determine the mean station waiting times for models with 3 or more stations. We show that some approximate models which have been proposed in the literature for providing upper bounds on the mean station waiting times do not always yield upper bounds. Extensive numerical tests indicate that a simple average of the two approximation schemes yields a close estimate of the true mean station waiting time. This average-of-approximations procedure appears to be robust for a large range of parameter values.Research supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP0045904.Research supported in part by the National Science Foundation under grant DMI-9500471.     

Sojourn times in a processor sharing queue with multiple vacations

We study an M/G/1 processor sharing queue with multiple vacations. The server only takes a vacation when the system has become empty. If he finds the system still empty upon return, he takes another vacation, and so on. Successive vacations are identically distributed, with a general distribution. When the service requirements are exponentially distributed we determine the sojourn time distribution of an arbitrary customer. We also show how the same approach can be used to determine the sojourn time distribution in an M/M/1-PS queue of a polling model, under the following constraints: the service discipline at that queue is exhaustive service, the service discipline at each of the other queues satisfies a so-called branching property, and the arrival processes at the various queues are independent Poisson processes. For a general service requirement distribution we investigate both the vacation queue and the polling model, restricting ourselves to the mean sojourn time.     

A Communication Multiplexer Problem: Two Alternating Queues with Dependent Randomly-Timed Gated Regime

Two random traffic streams are competing for the service time of a single server (multiplexer). The streams form two queues, primary (queue 1) and secondary (queue 0). The primary queue is served exhaustively, after which the server switches over to queue 0. The duration of time the server resides in the secondary queue is determined by the dynamic evolution in queue 1. If there is an arrival to queue 1 while the server is still working in queue 0, the latter is immediately gated, and the server completes service there only to the gated jobs, upon which it switches back to the primary queue. We formulate this system as a two-queue polling model with a single alternating server and with randomly-timed gated (RTG) service discipline in queue 0, where the timer there depends on the arrival stream to the primary queue. We derive Laplace–Stieltjes transforms and generating functions for various key variables and calculate numerous performance measures such as mean queue sizes at polling instants and at an arbitrary moment, mean busy period duration and mean cycle time length, expected number of messages transmitted during a busy period and mean waiting times. Finally, we present graphs of numerical results comparing the mean waiting times in the two queues as functions of the relative loads, showing the effect of the RTG regime.     

Stability of a two-queue cyclic polling system with BMAPs under gated service and state-dependent time-limited service disciplines

The stability of a cyclic polling system, with a single server and two infinite-buffer queues, is considered. Customers arrive at the two queues according to independent batch Markovian arrival processes. The first queue is served according to the gated service discipline, and the second queue is served according to a state-dependent time-limited service discipline with the preemptive repeat-different property. The state dependence is that, during each cycle, the predetermined limited time of the server’s visit to the second queue depends on the queue length of the first queue at the instant when the server last departed from the first queue. The mean of the predetermined limited time for the second queue either decreases or remains the same as the queue length of the first queue increases. Due to the two service disciplines, the customers in the first queue have higher service priority than the ones in the second queue, and the service fairness of the customers with different service priority levels is also considered. In addition, the switchover times for the server traveling between the two queues are considered, and their means are both positive as well as finite. First, based on two embedded Markov chains at the cycle beginning instants, the sufficient and necessary condition for the stability of the cyclic polling system is obtained. Then, the calculation methods for the variables related to the stability condition are given. Finally, the influence of some parameters on the stability condition of the cyclic polling system is analyzed. The results are useful for engineers not only checking whether the given cyclic polling system is stable, but also adjusting some parameters to make the system satisfy some requirements under the condition that the system is stable.     

Analysis of discrete time queues via the reversed process

A class of discrete-timeM/G/1 queues, including both round robin and last come first served service, in which customers are subject to permutations is considered. These time slotted queues, analogous to the symmetric queues of Kelly, are analyzed by examination of the time reversed process. Product form stationary distributions are found for a type of doubly stochastic server of Schassberger [5] and for a Bernoulli arrival process queue model of Henderson and Taylor [2].     

Ergodicity and analysis of the process describing the system state in polling systems with two queues

Consider a polling system of two queues served by a single server that visits the queues in cyclic order. The polling discipline in each queue is of exhaustive-type, and zero-switchover times are considered. We assume that the arrival times in each queue form a Poisson process and that the service times form sequences of independent and identically distributed random variables, except for the service distribution of the first customer who is served at each polling instant (the time in which the server moves from one queue to the other one). The sufficient and necessary conditions for the ergodicity of such polling system are established as well as the stationary distribution for the continuous-time process describing the state of the system. The proofs rely on the combination of three embedded processes that were previously used in the literature. An important result is that ρ=1 can imply ergodicity in one specific case, where ρ is the typical traffic intensity for polling systems, and ρ<1 is the classical non-saturation condition.     

A finite capacity multi-queueing system with priorities and with repeated server vacations

In this paper, we study an M/G/1 multi-queueing system consisting ofM finite capacity queues, at which customers arrive according to independent Poisson processes. The customers require service times according to a queue-dependent general distribution. Each queue has a different priority. The queues are attended by a single server according to their priority and are served in a non-preemptive way. If there are no customers present, the server takes repeated vacations. The length of each vacation is a random variable with a general distribution function. We derive steady state formulas for the queue length distribution and the Laplace transform of the queueing time distribution for each queue.     

A polling model with an autonomous server

This paper considers polling systems with an autonomous server that remains at a queue for an exponential amount of time before moving to a next queue incurring a generally distributed switch-over time. The server remains at a queue until the exponential visit time expires, also when the queue becomes empty. If the queue is not empty when the visit time expires, service is preempted upon server departure, and repeated when the server returns to the queue. The paper first presents a necessary and sufficient condition for stability, and subsequently analyzes the joint queue-length distribution via an embedded Markov chain approach. As the autonomous exponential visit times may seem to result in a system that closely resembles a system of independent queues, we explicitly investigate the approximation of our system via a system of independent vacation queues. This approximation is accurate for short visit times only.      

Symmetric queues served in cyclic order

Consider a symmetrical system of n queues served in cyclic order by a single server. It is shown that the stationary number of customers in the system is distributed as the sum of three independent random variables, one being the stationary number of customers in a standard M/G/1 queue. This fact is used to establish an upper bound for the mean waiting time for the case where at most k customers are served at each queue per visit by the server. This approach is also used to rederive the mean waiting times for the cases of exhaustive service, gated service, and serve at most one customer at each queue per visit by the server.     

Mixed gated/exhaustive service in a polling model with priorities

In this paper we consider a single-server polling system with switch-over times. We introduce a new service discipline, mixed gated/exhaustive service, that can be used for queues with two types of customers: high and low priority customers. At the beginning of a visit of the server to such a queue, a gate is set behind all customers. High priority customers receive priority in the sense that they are always served before any low priority customers. But high priority customers have a second advantage over low priority customers. Low priority customers are served according to the gated service discipline, i.e. only customers standing in front of the gate are served during this visit. In contrast, high priority customers arriving during the visit period of the queue are allowed to pass the gate and all low priority customers before the gate. We study the cycle time distribution, the waiting time distributions for each customer type, the joint queue length distribution of all priority classes at all queues at polling epochs, and the steady-state marginal queue length distributions for each customer type. Through numerical examples we illustrate that the mixed gated/exhaustive service discipline can significantly decrease waiting times of high priority jobs. In many cases there is a minimal negative impact on the waiting times of low priority customers but, remarkably, it turns out that in polling systems with larger switch-over times there can be even a positive impact on the waiting times of low priority customers.     

A queueing network with a single cyclically roving server

A queueingnetwork that is served by asingle server in a cyclic order is analyzed in this paper. Customers arrive at the queues from outside the network according to independent Poisson processes. Upon completion of his service, a customer mayleave the network, berouted to another queue in the network orrejoin the same queue for another portion of service. The single server moves through the different queues of the network in a cyclic manner. Whenever the server arrives at a queue (polls the queue), he serves the waiting customers in that queue according to some service discipline. Both the gated and the exhaustive disciplines are considered. When moving from one queue to the next queue, the server incurs a switch-over period. This queueing network model has many applications in communication, computer, robotics and manufacturing systems. Examples include token rings, single-processor multi-task systems and others. For this model, we derive the generating function and the expected number of customers present in the network queues at arbitrary epochs, and compute the expected values of the delays observed by the customers. In addition, we derive the expected delay of customers that follow a specific route in the network, and we introduce pseudo-conservation laws for this network of queues.Summary of notation B_i, B _i ^* (s) service time of a customer at queue i and its LST - b_i, b_i ⁽²⁾ mean and second moment of B_i - R_i, R _i ^* (s) duration of switch-over period from queue i and its LST - r_i, r_i mean and second moment of R_i - r, r⁽²⁾ mean and second moment of _i ^N =₁R_i - _i external arrival rate of type-i customers - _i total arrival rate into queue i - _i utilization of queue i; _i=_i - system utilization _i ^N =_1i - c=E[C] the expected cycle length - X _i ^j number of customers in queue j when queue i is polled - X_i=X _i ⁱ number of customers residing in queue i when it is polled - f_i(j) - X _i ^* number of customers residing in queue i at an arbitrary moment - Y_i the duration of a service period of queue i - W_i,T_i the waiting time and sojourn time of an arbitary customer at queue i - F^*(z₁, z₂,..., z_N) GF of number of customers present at the queues at arbitrary moments - F_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at polling instants of queue i - ¯F_i(z₁, z₂,...,z_N) GF of number of customers present at the queues at switching instants of queue i - V_i(z₁, z₂,..., z_N) GF of number of customers present at the queues at service initiation instants at queue i - ¯V_i(z₁,z₂,...,z_N) GF of number of customers present at the queues at service completion instants at queue i The work of this author was supported by the Bernstein Fund for the Promotion of Research and by the Fund for the Promotion of Research at the Technion.Part of this work was done while H. Levy was with AT&T Bell Laboratories.     

A single server queue with service interruptions

In this paper we characterize the queue-length distribution as well as the waiting time distribution of a single-server queue which is subject to service interruptions. Such queues arise naturally in computer and communication problems in which customers belong to different classes and share a common server under some complicated service discipline. In such queues, the viewpoint of a given class of customers is that the server is not available for providing service some of the time, because it is busy serving customers from a different class. A natural special case of these queues is the class of preemptive priority queues. In this paper, we consider arrivals according the Markovian Arrival Process (MAP) and the server is not available for service at certain times. The service times are assumed to have a general distribution. We provide numerical examples to show that our methods are computationally feasible. This revised version was published online in June 2006 with corrections to the Cover Date.     

Equilibrium customers’ choice between FCFS and random servers

Consider two servers of equal service capacity, one serving in a first-come first-served order (FCFS), and the other serving its queue in random order. Customers arrive as a Poisson process and each arriving customer observes the length of the two queues and then chooses to join the queue that minimizes its expected queueing time. Assuming exponentially distributed service times, we numerically compute a Nash equilibrium in this system, and investigate the question of which server attracts the greater share of customers. If customers who arrive to find both queues empty independently choose to join each queue with probability 0.5, then we show that the server with FCFS discipline obtains a slightly greater share of the market. However, if such customers always join the same queue (say of the server with FCFS discipline) then that server attracts the greater share of customers. This research was supported by the Israel Science Foundation grant No. 526/08.     

A two-queue model with Bernoulli service schedule and switching times

In this paper, we present a detailed analysis of a cyclic-service queueing system consisting of two parallel queues, and a single server. The server serves the two queues with a Bernoulli service schedule described as follows. At the beginning of each visit to a queue, the server always serves a customer. At each epoch of service completion in the ith queue at which the queue is not empty, the server makes a random decision: with probability p_i, it serves the next customer; with probability 1-p_i, it switches to the other queue. The server takes switching times in its transition from one queue to the other. We derive the generating functions of the joint stationary queue-length distribution at service completion instants, by using the approach of the boundary value problem for complex variables. We also determine the Laplace-Stieltjes transforms of waiting time distributions for both queues, and obtain their mean waiting times. This revised version was published online in June 2006 with corrections to the Cover Date.     

A single server queue with gated processor-sharing discipline

In this paper we consider a single server queue in which arrivals occur according to a Poisson process and each customer's service time is exponentially distributed. The server works according to the gated process-sharing discipline. In this discipline, the server provides service to a batch of at mostm customers at a time. Once a batch of customers begins service, no other waiting customer can receive service until all members of the batch have completed their service. For this queue, we derive performance characteristics, such as waiting time distribution, queue length distribution etc. For this queue, it is possible to obtain the mean conditional response time for a customer whose service time is known. This conditional response time is a nonlinear function (as opposed to the linear case for the ordinary processor-sharing queue). A special case of the queue (wherem=) has an interesting and unusual solution. For this special case, the size of the batch for service is a Markov chain whose steady state distribution can be explicitly written down. Apart from the contribution to the theory of Markov chains and queues, the model may be applicable to scheduling of computer and communication systems.     

Fluid polling systems

We study N-queues single-server fluid polling systems, where a fluid is continuously flowing into the queues at queue-dependent rates. When visiting and serving a queue, the server reduces the amount of fluid in the queue at a queue-dependent rate. Switching from queue i to queue j requires two random-duration steps: (i) departing queue i, and (ii) reaching queue j. The length of time the server resides in a queue depends on the service regime. We consider three main regimes: Exhaustive, Gated, and Globally-Gated. Two polling procedures are analyzed: (i) cyclic and (ii) probabilistic. Under steady-state, we derive the Laplace–Stieltjes transform (LST), mean, and second moment of the amount of flow at each queue at polling instants, as well as at an arbitrary moment. We further calculate the LST and mean of the “waiting time” of a drop at each queue and derive expressions for the mean total load in the system for the various service regimes. Finally, we explore optimal switching procedures.