Joint Queue Length Distribution Research Articles

Queue lengths and workloads in polling systems

For a single-server multi-station polling system, we focus on the generating function and Laplace–Stieltjes transform of the time-stationary joint queue length and workload distributions, respectively, under no further assumptions on the service discipline. We express these quantities as expressions involving the generating functions of the joint queue length distribution at visit beginnings and visit completions at the various stations. The latter is known for a broad variety of cases. Finally, we identify a workload decomposition result.

A preemptive resume priority retrial queue with state dependent arrivals, unreliable server and negative customers

In this paper we consider an unreliable single server retrial queue accepting two types of customers, with negative arrivals, preemptive resume priorities and vacations. A distinguishing feature of the model is that the rates of the Poisson arrival process depends on the server state. For this model we investigate the stability conditions and the joint queue length distribution in steady state. We also prove that our model satisfies the stochastic decomposition property. Transient, as well as steady state solutions for reliability measures are obtained. Finally, numerical results demonstrate the typical features of the model under consideration.

On the stationary LCFS-PR single-server queue: A characterization via stochastic intensity

We consider a stationary single-server queue with preemptive-resumelast-come, first-served (LCFS-PR) queueing discipline.The LCFS-PR single-server queue has some interesting properties andhas been studied in the literature.In this paper, we generalize the previous results such that the inputprocess to the queue is given as a general stationary marked pointprocess and derive some formulas concerning the joint distribution ofqueue length and remaining service times of respective customers inthe system at arbitrary time instances as well as at arrivalinstances.The tool for derivation is the Palm-martingale calculus; that is, theconnection between the notion of Palm probability and that ofstochastic intensity.

Analysis of the M/G/1 queue with discriminatory random order service policy

We consider an M/G/1 queue with different classes of customers and discriminatory random order service (DROS) discipline. The DROS discipline generalizes the random order service (ROS) discipline: when the server selects a customer to serve, all customers waiting in the system have the same selection probability under ROS discipline, whereas customers belonging to different classes may have different selection probabilities under DROS discipline. For the M/G/1 queue with DROS discipline, we derive equations for the joint queue length distributions and for the waiting time distributions of each class. We also obtain the moments of the queue lengths and the waiting time of each class. Numerical results are given to illustrate our results.

A polling model with smart customers

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 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.

A polling model with multiple priority levels

In this paper we consider a single-server cyclic polling system. Between visits to successive queues, the server is delayed by a random switch-over time. The order in which customers are served in each queue is determined by a priority level that is assigned to each customer at his arrival. For this situation the following service disciplines are considered: gated, exhaustive, and globally gated. We study the cycle time distribution, the waiting times 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.

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.

A Two-Queue Polling Model with Two Priority Levels in the First Queue

In this paper we consider a single-server cyclic polling system consisting of two queues. Between visits to successive queues, the server is delayed by a random switch-over time. Two types of customers arrive at the first queue: high and low priority customers. For this situation the following service disciplines are considered: gated, globally gated, and exhaustive. We study the cycle time distribution, the waiting times for each customer type, the joint queue length distribution at polling epochs, and the steady-state marginal queue length distributions for each customer type.

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 performance analysis of discrete-time tandem queues with Markovian sources

We present an exact performance analysis of tandem networks with an arbitrary number of multiplexers. Each multiplexer is fed by the output of its upstream neighbor as well as the traffic generated by a number of independent binary Markovian sources. We model the tandem network as a discrete-time queueing system. After determining the unknown boundary functions, the probability generating function (PGF) of the joint distribution of queue length and number of O n sources for each multiplexer is derived as a function of the transform of the busy period of its upstream neighbor. From this PGF, we determine closed form expressions for the mean and variance of queue length, as well as the mean packet delay. The solution has also been extended to tandem networks in which each multiplexer is fed by multiple types of traffic. Then, numerical results are presented and it has been shown that the multiplexing smooths the traffic. The mean queue lengths are very easy to calculate and the computation complexity does not increase with the number of sources in the network.

Upper bound for the decay rate of the joint queue-length distribution in a two-node Markovian queueing system

This paper studies the geometric decay property of the joint queue-length distribution {p(n 1,n 2)} of a two-node Markovian queueing system in the steady state. For arbitrarily given positive integers c 1,c 2,d 1 and d 2, an upper bound $\overline{\eta}(c_{1},c_{2})$ of the decay rate is derived in the sense $$\exp\Bigl\{\limsup_{n\rightarrow\infty}n^{-1}\log p(c_{1}n+d_{1},c_{2}n+d_{2})\Bigr\}\leq\overline{\eta}(c_{1},c_{2})<1.$$ It is shown that the upper bound coincides with the exact decay rate in most systems for which the exact decay rate is known. Moreover, as a function of c 1 and c 2, $\overline{\eta }(c_{1},c_{2})$ takes one of eight types, and the types explain some curious properties reported in Fujimoto and Takahashi (J. Oper. Res. Soc. Jpn. 39:525---540 [1996]).

Discriminatory Processor Sharing Queues and the DREB Method

We study a discriminatory processor-sharing (DPS) queue with a Markovian arrival process (MAP) and K classes of customers. This system can be modeled into a K-dimensional quasi-birth-and death (QBD) process by the standard matrix-analytical approach, but such a process is computationally difficult to handle. We present a different formulation by which the K-dimensional QBD process is reduced to a single-dimensional level-dependent QBD process with expanding blocks. We derive analytical expressions and use them to compute the sojourn time and joint queue length distributions efficiently. This method allows us to study numerically a wide range of important open queueing problems that are common in computer and communication systems, such as various priority queues, queueing networks, and other dependent queues.

Asymptotic regimes and approximations for discriminatory processor sharing

We study the joint queue length distribution of the Discriminatory Processor Sharing model, assuming all classes have phase-type service requirement distributions. We show that the moments of the joint queue length distribution can be obtained by solving linear equations. We use this to study the system in two asymptotic regimes. In the first regime, the different user classes operate on strictly separated time scales. Then we study the system in heavy traffic.

Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams

This paper considers a work-conserving FIFO single-server queue with multiple batch Markovian arrival streams governed by a continuous-time finite-state Markov chain. A particular feature of this queue is that service time distributions of customers may be different for different arrival streams. After briefly discussing the actual waiting time distributions of customers from respective arrival streams, we derive a formula for the vector generating function of the time-average joint queue length distribution in terms of the virtual waiting time distribution. Further assuming the discrete phase-type batch size distributions, we develop a numerically feasible procedure to compute the joint queue length distribution. Some numerical examples are provided also.

Order Fill Rate, Leadtime Variability, and Advance Demand Information in an Assemble-to-Order System

We study an assemble-to-order system with stochastic leadtimes for component replenishment. There are multiple product types, of which orders arrive at the system following batch Poisson processes. Base-stock policies are used to control component inventories. We analyze the system as a set of queues driven by a common, multiclass batch Poisson input, and derive the joint queue-length distribution. The result leads to simple, closed-form expressions of the first two moments, in particular the covariances, which capture the dependence structure of the system. Based on the joint distribution and the moments, we derive easy-to-compute approximations and bounds for the order fulfillment performance measures. We also examine the impact of demand and leadtime variability, and investigate the value of advance demand information.

A Dynamic Priority Queue Model for Simultaneous Service of Two Traffic Types

We consider a priority queue with a dynamic, queue-length-threshold scheduling policy. Customers are classed into two types (type-1 and type-2), and the service order of the two classes depends on the queue length of the type-1 queue. The high priority (type-2) class (e.g., voice) is served until the low priority (e.g., data) queue exceeds the threshold L, at which time service is given to the low priority class until its queue length decreases to L. The arrivals of the two classes follow independent Poisson processes, and the service time of each customer has an exponential distribution with parameter $\mu$. We derive the balance equations in the steady state, and explicitly obtain the joint probability generating function for the queue lengths of the two customer classes. This gives the joint queue length distribution as an integral. We then obtain detailed asymptotic results for the joint distribution. In particular, we study the tail behavior. We also discuss heavy traffic diffusion approximations for this model.

The M/G/1 FIFO queue with several customer classes

In this note we present short derivations of the joint queue length distribution in the M/G/1 queue with several classes of customers and FIFO service discipline.

STATIONARY QUEUE LENGTH IN A FIFO SINGLE SERVER QUEUE WITH SERVICE INTERRUPTIONS AND MULTIPLE BATCH MARKOVIAN ARRIVAL STREAMS

This paper considers a FIFO single-server queue with service interruptions and multiple batch Markovian arrival streams. The server state (on and off), the type of arriving customers and their batch size are assumed to be governed by a continuous-time Markov chain with finite states. To put it more concretely, the marginal process of the server state is a phase-type alternating Markov renewal process, the marginal arrival process is a batch marked Markovian arrival process, and they may be correlated Further, service times of arriving customers are allowed to depend on both their arrival stream and the server state on arrival. For such a queue, we derive the vector joint generating function of the numbers of customers from respective arrival streams. Further assuming discrete phase-type batch size distributions, we establish a numerical algorithm to compute the joint queue length distribution at a random point in time. Finally, we show some numerical examples and examine the impact of system parameters on the queue length distribution

A two-queue and two-server model with a threshold-based control service policy

This paper investigates a queueing system consisting of two-parallel queues and two servers. The service policy is a threshold-based control one such that two thresholds v (⩾0) and N (⩾v) called control level are set up in one of the two queues, say, the second queue. At each epoch of service completion, the server decides which queue is to be served next according to the control level the number of customers in the second queue reaches. For both queues, the arrival processes are Poisson, and the service times are exponentially distributed with different means. We derive the generating functions of the stationary joint queue-length distribution, and then obtain the mean queue length and the mean waiting time for each queue.

Monotonicity and Asymptotic Queue-Length Distribution in Discrete-Time Networks

For closed, cyclic, discrete-time networks with one server per node and with independent, geometric service times, in equilibrium, the joint queue-length distribution can be realized as the joint distribution of independent random variables, conditionally given their sum. This tool helps establish monotonicity properties of performance measures and also helps show that the queue-length random variables are negatively associated. The queue length at a node is asymptotically analyzed through a family of networks with a fixed number of node types, where the number of nodes approaches infinity, the ratio of jobs to nodes has a positive limit, and each node type has a limiting density. The queue-length distribution at any node is shown to converge, in a strong sense, to a distribution that is conditionally geometric. As a by-product, this approach settles open issues regarding occupancy proportion and average queue length at a node type.