Two queues with vastly different arrival rates and processor-sharing factors

We consider a 2-class queueing system, operating under a generalized processor-sharing discipline. The arrival rate to the secondary queue is much smaller than that to the primary queue, while the exponentially distributed service requirements have comparable parameters. The primary queue is assumed to be heavily loaded, so the processor-sharing factor for the secondary queue is assumed to be relatively small. We use singular perturbation analyses in a small parameter measuring the ratio of arrival rates, and the closeness of the system to instability. Two different regimes are analyzed, corresponding to a heavily loaded and a lightly loaded secondary queue, respectively. With suitable scaling of variables, lowest order asymptotic approximations to the joint stationary distribution of the numbers of jobs in the two queues are derived, as well as to the marginal distributions.     

Processor sharing for two queues with vastly different rates

We consider a 2-class queueing system, operating under a generalized processor-sharing discipline, in an asymptotic regime where the arrival and service rates of the two classes are vastly different. We use regular and singular perturbation analyses in a small parameter measuring this difference in rates. It is assumed that the system is stable, and not close to instability. Three different regimes are analyzed, corresponding to an underloaded, an overloaded and a critically loaded fast queue, respectively. In the first two regimes the lowest order approximation to the joint stationary distribution of the queue lengths is derived. For a critically loaded fast queue only the mean queue lengths are investigated, and the asymptotic matching, to lowest order, with the results for an underloaded and an overloaded fast queue is established.      

Interacting queues in heavy traffic

We consider a system of parallel queues with Poisson arrivals and exponentially distributed service requirements. The various queues are coupled through their service rates, causing a complex dynamic interaction. Specifically, the system consists of one primary queue and several secondary queues whose service rates depend on whether the primary queue is empty or not. Conversely, the service rate of the primary queue depends on which of the secondary queues are empty.     

Joint distributions in Poissonian tandem queues

A pair of single server queues arranged in series is considered. The input flow is Poisson and service times are mutually independent and exponentially distributed in each station. The joint distributions of the stationary waiting times and queue lengths at the arrival epoch are treated.     

Asymptotic independence of queues under randomized load balancing

Randomized load balancing greatly improves the sharing of resources while being simple to implement. In one such model, jobs arrive according to a rate-??N Poisson process, with ??<1, in a system of N rate-1 exponential server queues. In Vvedenskaya et al. (Probl. Inf. Transm. 32:15?C29, 1996), it was shown that when each arriving job is assigned to the shortest of D, D??2, randomly chosen queues, the equilibrium queue sizes decay doubly exponentially in the limit as N????. This is a substantial improvement over the case D=1, where queue sizes decay exponentially. The reasoning in Vvedenskaya et al. (Probl. Inf. Transm. 32:15?C29, 1996) does not easily generalize to jobs with nonexponential service time distributions. A?modularized program for treating randomized load balancing problems with general service time distributions was introduced in Bramson et al. (Proc. ACM SIGMETRICS, pp.?275?C286, 2010). The program relies on an ansatz that asserts that, for a randomized load balancing scheme in equilibrium, any fixed number of queues become independent of one another as N????. This allows computation of queue size distributions and other performance measures of interest. In this article, we demonstrate the ansatz in several settings. We consider the least loaded balancing problem, where an arriving job is assigned to the queue with the smallest workload. We also consider the more difficult problem, where an arriving job is assigned to the queue with the fewest jobs, and demonstrate the ansatz when the service discipline is FIFO and the service time distribution has a decreasing hazard rate. Last, we show the ansatz always holds for a sufficiently small arrival rate, as long as the service distribution has 2 moments.     

Log-weight scheduling in switched networks

We consider switched queueing networks in which there are constraints on which queues may be served simultaneously. The scheduling policy for such a network specifies which queues to serve at any point in time. We introduce and study a variant of the popular maximum weight or backpressure policy which chooses the collection of queues to serve that has maximum weight. Unlike the maximum weight policies studied in the literature, the weight of a queue depends on logarithm of its queue-size in this paper. For any multihop switched network operating under such maximum log-weighted policy, we establish that the network Markov process is positive recurrent as long as it is underloaded. As the main result of this paper, a meaningful fluid model is established as the formal functional law of large numbers approximation. The fluid model is shown to be work-conserving. That is, work (or total queue-size) is nonincreasing as long as the network is underloaded or critically loaded. We identify invariant states or fixed points of the fluid model. When underloaded, null state is the unique invariant state. For a critically loaded fluid model, the space of invariant states is characterized as the solution space of an optimization problem whose objective is lexicographic ordering of total queue-size and the negative entropy of the queue state. An important contribution of this work is in overcoming the challenge presented by the log-weight function in establishing meaningful fluid model. Specifically, the known approaches in the literature primarily relied on the “scale invariance” property of the weight function that log-function does not possess.     

Stationary tail probabilities in exponential server tandems with renewal arrivals

The problem considered is that of estimating the tail stationary probability for two exponential server queues in series fed by renewal arrivals. We compute the tail of the marginal queue length distribution at the second queue. The marginal at the first queue is known by the classical result for the GI/M/1 queue. The approach involves deriving necessary and sufficient conditions on the paths of the arrival and virtual service processes in order to get a large queue size at the second queue. We then use large deviations estimates of the probabilities of these paths, and solve a constrained convex optimization problem to find the most likely path leading to a large queue size. We find that the stationary queue length distribution at the second queue has an exponentially decaying tail, and obtain the exact rate of decay.Research supported in part by NSF grant NCR 88-57731 and the AT & T Foundation.     

Optimal stochastic scheduling in a single server biclass retrial queueing system

The paper deals with the assignment of a single server to two retrial queues. Each customer reapplies for service after an exponentially distributed amount of time. The server operates at customer dependent exponential rates. There are holding costs and costs during service per customer and per unit of time. We provide conditions on which it is optimal to allocate the server to queue 1 or 2 in order to minimize the expected total costs until the system is cleared.     

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.     

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.     

Analysis of the asymmetric shortest queue problem

In this paper we study a system consisting of two parallel servers withdifferent service rates. Jobs arrive according to a Poisson stream and generate an exponentially distributed workload. On arrival a job joins the shortest queue and in case both queues have equal lengths, he joins the first queue with probabilityq and the second one with probability 1 −q, whereq is an arbitrary number between 0 and 1. In a previous paper we showed for the symmetric problem, that is for equal service rates andq = 1/2, that the equilibrium distribution of the lengths of the two queues can be exactly represented by an infinite sum of product form solutions by using an elementary compensation procedure. The main purpose of the present paper is to prove a similar product form result for the asymmetric problem by using a generalization of the compensation procedure. Furthermore, it is shown that the product form representation leads to a numerically efficient algorithm. Essentially, the method exploits the convergence properties of the series of product forms. Because of the fast convergence an efficient method is obtained with upper and lower bounds for the exact solution. For states further away from the origin the convergence is faster. This aspect is also exploited in the paper.     

Hierarchical Queue Networks with Partially Shared Servicing

The queue network studied consists of n infinite queues in parallel served by independent servers and by other servers all linked to form a hierarchical structure. The total service a unit receives depends partially on other units in service. We call this type of servicing partially shared servicing. All interarrival times as well as service times are assumed exponentially distributed. The characteristic of interest is the traffic intensity of the infinite queues. Some simple formulae are obtained. An application to modelling a disc I/O system is described. The model turns out to be useful and accurate with wide applicability.     

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.     

Waiting time analysis of the multiple priority dual queue with a preemptive priority service discipline

The dual queue consists of two queues, called the primary queue and the secondary queue. There is a single server in the primary queue but the secondary queue has no service facility and only serves as a holding queue for the overloaded primary queue. The dual queue has the additional feature of a priority scheme to help reduce congestion. Two classes of customers, class 1 and 2, arrive to the dual queue as two independent Poisson processes and the single server in the primary queue dispenses an exponentially distributed service time at the rate which is dependent on the customer’s class. The service discipline is preemptive priority with priority given to class 1 over class 2 customers. In this paper, we use matrix-analytic method to construct the infinitesimal generator of the system and also to provide a detailed analysis of the expected waiting time of each class of customers in both queues.     

A scheduling problem for several parallel servers

In this paper, we study a scheduling problem of jobs from two different queues on several parallel servers. Jobs have exponentially distributed processing times, and incur costs per unit of time, until they leave the system, and there are no arrivals to the system at any time. The objective is to find the optimal strategy, i.e., to allocate the servers to the queues, such that the expected holding costs are minimized. We give a sufficient condition for which it is always optimal to allocate the servers only to jobs of a certain queue. Finally, the case of two servers is completely solved.     

On the stability of open networks: A unified approach by stochastic dominance

Using stochastic dominance, in this paper we provide a new characterization of point processes. This characterization leads to a unified proof for various stability results of open Jackson networks where service times are i.i.d. with a general distribution, external interarrivai times are i.i.d. with a general distribution and the routing is Bernoulli. We show that if the traffic condition is satisfied, i.e., the input rate is smaller than the service rate at each queue, then the queue length process (the number of customers at each queue) is tight. Under the traffic condition, the pth moment of the queue length process is bounded for allt if the p+lth moment of the service times at all queues are finite. If, furthermore, the moment generating functions of the service times at all queues exist, then all the moments of the queue length process are bounded for allt. When the interarrivai times are unbounded and non-lattice (resp. spreadout), the queue lengths and the remaining service times converge in distribution (resp. in total variation) to a steady state. Also, the moments converge if the corresponding moment conditions are satisfied.     

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.     

Stability and delay of distributed scheduling algorithms for networks of conflicting queues

This paper explains recent results on distributed algorithms for networks of conflicting queues. At any given time, only specific subsets of queues can be served simultaneously. The challenge is to select the subsets in a distributed way to stabilize the queues whenever the arrival rates are feasible. One key idea is to formulate the subset selection as an optimization problem where the objective function includes the entropy of the distribution of the selected subsets. The dual algorithm for solving this optimization problem provides a distributed scheduling algorithm that requires only local queue-length information. The algorithm is based on the CSMA (Carrier Sense Multiple Access) protocol in wireless networks. We also explain recent results, some of them unpublished so far, on the delay properties of these algorithms. In particular, we present a framework for queuing stability under bounded CSMA parameters, and show how the expected queue lengths depend on the throughput region to be supported. When the arrival rates are within a fraction of the capacity region, queue lengths that are polynomial (or even logarithmic) in the number of queues can be achieved.     

An assignment problem for a parallel queueing system with two heterogeneous servers

In this paper, we consider an optimization problem for a parallel queueing system with two heterogeneous servers. Each server has its own queue and customers arrive at each queue according to independent Poisson processes. Each service time is independent and exponentially distributed. When a customer arrives at queue 1, the customers in queue 1 can be transferred to queue 2 by paying an assignment cost which is proportional to the number of moved customers. Holding cost is a function of the pair of queue lengths of the two servers. Our objective is to minimize the expected total discounted cost. We use the dynamic programming approach for this problem. Considering the pair of queue lengths as a state space, we show that the optimal policy has a switch over structure under some conditions on the holding cost.