Probability Distribution Of Queue Length Research Articles

Modeling the dynamical behavior of the passenger-taxi queue matching problem at traffic hubs

Abstract Taxis are one of the most important modes of transport in major passenger traffic hubs. Due to the inherent unpredictability of passenger arrivals and their strong correlation with passenger arrival times, it often results in excessively long passenger queues if a large number of passengers arrive suddenly, or excessively long taxi queues at the other end if a few passengers arrive during the non-burst period at the taxi stand in the traffic hubs. In particular, when a sudden arrival of a large number of passengers fails to evacuate in a short period of time, followed by another burst in passenger arrivals, the service performance of the system deteriorates dramatically. In order to quantitatively analyze the dynamical queueing behavior of the passenger-taxi matching problem at transport hubs against the background of many uncertain factors, we propose a double-input matching model based on queueing theory, which covers basic practical elements, including time-varying arrivals of passengers and taxis, a randomly matched number of passengers, a random matching time, and multiple waiting queues. We determine the steady-state condition of the system, derive the steady-state queue length probability distributions of passengers and taxis, and further obtain the overall average queueing performance metrics of passengers and taxis and the dynamic queueing metrics over time. Numerical simulations examine the impact of the stochastic arrival process of passengers over a long period of time on the dynamical performance metrics of the system. In particular, for continuous and discontinuous bursty arrivals of passengers over a period of time, we clarify how the long queues caused by bursty arrivals of passengers dissipate at subsequent times and also examine the impact of variations in passenger arrival rates during a specific time period on queueing performance throughout the system.

Continuous-type [formula omitted]-inventory model with an attached M/M/1 queue and lost sales

In this study, we consider an M/M/1 queuing model with an attached continuous-type inventory. Customers arrive in the system according to a Poisson process and are served individually on a first-come, first-served basis. The service times of customers are assumed to be independent and identically distributed exponential random variables. Along with the queue, there is an internal finite storage for the inventory and each service requires an exponentially distributed random amount H of inventory from the storage. Therefore, a customer leaves the system with H amount of item at his/her service completion time. The inventory is replenished by an outside supplier with a random lead time under an (s,Q) inventory control policy. We assume that the customers who arrive during stock-out periods are lost (lost sales). For this queuing-inventory system, we derive the stationary joint probability distribution of queue length and inventory level in explicit product form. Numerical examples followed by a cost model are also presented.

Maximum Entropy Analysis to the N Policy M/G/1 Queue with Working Breakdowns

Abstract This paper deals with the N policy M/G/1 queue with working breakdowns. In this queueing system, the steady-state probabilities cannot be derived explicitly. Thus, we employ a maximum entropy approach with several constraints to develop the approximate formulae for the steady-state probability distributions of queue length and the expected waiting time in the queue. We perform a comparative analysis between the approximate results with established exact results for different service time distributions, such as exponential, two-stage Erlang, two-stage hyper-exponential, and deterministic. Numerical results demonstrate that the maximum entropy approach is quite accurate for practical purposes and is useful for complex queueing-systems solving.

A two-stage queueing network with MAP inputs and buffer sharing

In this paper, we study a two-stage tandem queueing network with MAP inputs and buffer sharing. The two stages share the same buffer. By using Markov process, we give an exact analysis of the queueing network. Since the customer arrival is not a Poisson process, the PASTA (Poisson Arrivals See Time Averages) property does not hold. A matrix filtration technique is proposed to derive the probability distribution of queue length at arrivals. Our objective is to investigate how the buffer sharing policy is mitigate the tradeoff between the probability that an arriving customer is lost and the probability that the first-stage server is blocked. The numerical results show that buffer sharing policy is more flexible, especially when the inputs have large variant and are correlated.

A Simple and Extended Computational Analysis of M/Gj(a,b)/1 and M/Gj(a,b)/1/(B + b) Queues Using Roots

A recent article on bulk-service queues gives a generating function for the steady-state probabilities of the embedded Markov chain for a single-server infinite-space system in which, the customers arrive according to a Poisson process and are served in batches with quorum a and capacity b, and the service time follows a general distribution dependent on batch size. This system is equivalent to the bulk queueing model M/Gj(a,b)/1, whose general solution requires finding the roots of the denominator of the underlying generating function. The article claims that the use of roots may result in numerical inaccuracies, especially for large values of b. Hence it only solves for the finite-space model M/Gj(a,b)/1/(B + b) using (B+1) simultaneous linear equations. We present a simple way to obtain the probability distribution of queue length at post-departure epochs for the infinite-space model M/Gj(a,b)/1 using roots, then an alternative method to solve the finite-space queue M/Gj(a,b)/1/(B+b). We derive, for the first time, closed-form formulas for the queue-length distribution of models with deterministic service time for both infinite (M/Dj(a,b)/1) and finite-space (M/Dj(a,b)/1/(B+b)) systems. We also show that the queue-length distribution of (M/Dj(a,b)/1 can be approximated by a Poisson distribution when the traffic intensity p is low. Numerical results are both tabulated and graphed.

Multi-server system with single working vacation

We consider an M/M/R queue with vacations, in which the server works with different service rates rather than completely terminates service during his vacation period. Service times during vacation period, service times during service period and vacation times are all exponentially distributed. Neuts’ matrix–geometric approach is utilized to develop the computable explicit formula for the probability distributions of queue length and other system characteristics. A cost model is derived to determine the optimal values of the number of servers and the working vacation rate simultaneously, in order to minimize the total expected cost per unit time. Under the optimal operating conditions, numerical results are provided in which several system characteristics are calculated based on assumed numerical values given to the system parameters.

DISCRETE-TIME QUEUE WITH VARIABLE SERVICE CAPACITY

This paper considers a discrete-time queueing system with variable service capacity. Using the supplementary variable method and the generating function technique, we compute the joint probability distribution of queue length and remaining service time at an arbitrary slot boundary, and also compute the distribution of the queue length at a departure time.

Flow equivalence and stochastic equivalence in G-networks

G-networks are novel product form queuing networks that, in addition to ordinary customers, contain unusual entities such as “negative customers” which eliminate normal customers, and “triggers” that move other customers from some queue to another. Recently we introduced one more special type of customer, a “reset”, which may be sent out by any server at the end of a service epoch, and that will reset the queue to which it arrives into its steady state when that queue is empty. A reset which arrives to a non-empty queue has no effect at all. The sample paths of a system with resets is significantly different from that of a system without resets, because the arrival of a reset to an empty queue will provoke a finite positive jump in queue length which may be arbitrarily large, while without resets positive jumps are only of size + 1 and they occur only when a positive customer arrives to a queue. In this paper we review this novel model, and then discuss its traffic equations. We introduce the concept of “stationary equivalence” for queueing models, and of “flow equivalence” for distinct queueing models. We show that the flow equivalence of two G-networks implies that they are also stationary equivalent. We then show that the stationary probability distribution of a G-network with resets is identical to that of a G-network without resets whose transition probabilities for positive (ordinary) customers has been increased in a specific manner. Our results show that a G-network with resets has the same form of traffic equations and the same joint stationary probability distribution of queue length as that of a G-network without resets.

A new blocking problem from Java-based schedulers

We consider a form of blocking, which is typical in client–server systems including those implemented under the Enterprise JavaBean (EJB) specification. The novel feature is that tasks must wait for one of a number of parallel queues to clear its outstanding work. Thus, blocking time is the minimum of sojourn times at the parallel queues. Under certain simplifying assumptions, we solve this model for the probability distribution of blocking time and obtain a simple formula for its mean value. We then use this result in an aggregate server model of a larger queueing network in which further non-standard techniques are included to represent this form of blocking. We compare our approximate results against simulation data, obtaining good agreement for both system throughput and queue length probability distributions at equilibrium.

Convergence of Prox-Regularization Methods for Generalized Fractional Programming

We propose an approximate analysis of open systems of tandem queues with blocking caused by finite buffers between servers. Our approach relies on the use of marginal probability distributions (“state equivalence”) coupled with an approximate evaluation of the conditional probabilities introduced through the equivalence. The method iterates over consecutive pairs of servers using the solution of a two-queue system as a building block. It produces performance measures for individual servers as well as an approximation to joint queue-length probability distributions for pairs of neighboring stations. Experience indicates that the number of iterations required for the method grows moderately with the number of nodes in the network. We give examples to demonstrate the accuracy and the convergence properties of the proposed approximation.

A Markov modulated multi-server queue with negative customers – The MM CPP/GE/c/L G-queue

We obtain the queue length probability distribution at equilibrium for a multi-server queue with generalised exponential service time distribution and either finite or infinite waiting room. This system is modulated by a continuous time Markov phase process. In each phase, the arrivals are a superposition of a positive and a negative arrival stream, each of which is a compound Poisson process with phase dependent parameters, i.e. a Poisson point process with bulk arrivals having geometrically distributed batch size. Such a queueing system is well suited to B-ISDN/ATM networks since it can account for both burstiness and correlation in traffic. The result is exact and is derived using the method of spectral expansion applied to the two dimensional (queue length by phase) Markov process that describes the dynamics of the system. Several variants of the system are considered, applicable to different modelling situations, such as server breakdowns, cell losses and load balancing. We also consider the departure process and derive its batch size distribution and the Laplace transform of the interdeparture time probability density function. From this, a recurrence formula is obtained for its moments. The analysis therefore provides the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes.

The MMCPP/GE/ c Queue

We obtain the queue length probability distribution at equilibrium for a multi-server, single queue with generalised exponential (GE) service time distribution and a Markov modulated compound Poisson arrival process (MMCPP) – i.e., a Poisson point process with bulk arrivals having geometrically distributed batch size whose parameters are modulated by a Markovian arrival phase process. This arrival process has been considered appropriate in ATM networks and the GE service times provide greater flexibility than the more conventionally assumed exponential distribution. The result is exact and is derived, for both infinite and finite capacity queues, using the method of spectral expansion applied to the two dimensional (queue length by phase of the arrival process) Markov process that describes the dynamics of the system. The Laplace transform of the interdeparture time probability density function is then obtained. The analysis therefore could provide the basis of a building block for modelling networks of switching nodes in terms of their internal arrival processes, which may be both correlated and bursty.

Control of arrivals in a finite buffered queue with setup costs

We consider finite buffered queues where the arrival process is controlled by shutting down and restarting the arrival stream. In the absence of holding costs for items in the queue, the optimal (s, S) policy can be characterised by relating the arrival control problem to a corresponding service control problem. With the inclusion of holding costs however, this characterisation is not valid and efficient numerical computation of the queue length probability distribution is necessary. We perform this computation by using a duality property which relates queue lengths in the controlled arrival system to a controlled service system. Numerical results which analyse the effect of setup and holding costs and the variability of the arrival process on the performance of the system are included.

The M/G/1 queue with negative customers

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

The M/G/1 queue with negative customers

We derive expressions for the generating function of the equilibrium queue length probability distribution in a single server queue with general service times and independent Poisson arrival streams of both ordinary, positive customers and negative customers which eliminate a positive customer if present. For the case of first come first served queueing discipline for the positive customers, we compare the killing strategies in which either the last customer in the queue or the one in service is removed by a negative customer. We then consider preemptive-restart with resampling last come first served queueing discipline for the positive customers, combined with the elimination of the customer in service by a negative customer—the case of elimination of the last customer yields an analysis similar to first come first served discipline for positive customers. The results show different generating functions in contrast to the case where service times are exponentially distributed. This is also reflected in the stability conditions. Incidently, this leads to a full study of the preemptive-restart with resampling last come first served case without negative customers. Finally, approaches to solving the Fredholm integral equation of the first kind which arises, for instance, in the first case are considered as well as an alternative iterative solution method.

An approximate analysis of asynchronous, packet-switched buffered banyan networks with blocking

An approximate algorithm is developed for the performance analysis of buffered, packet-switched, asynchronous networks with no feedback. We focus on banyan networks which are important in parallel computer architectures and telecommunication systems to illustrate our approach and our algorithm is tailored to this problem. The networks we consider are organised in a finite number of stages through each of which a task passes successively in its transmission. The components which forward messages in each stage are modelled independently as small sub-networks of queues; these components are crossbars in the case of banyan networks. Blocking can occur on several upstream paths preceding a full component and the inter-stage blocking effect is taken into account iteratively, blocking probabilities being computed in terms of results of a previous step. The networks we consider may be non-homogeneous in that different servers may have different rates and the arrival processes need not be identical. We derive the queue length probability distribution of each switch from which performance measures, such as blocking probability, mean buffer occupancy and mean transmission time, follow.

Transient analysis of a switched Poisson arrival queue under overload control

This paper studies the transient behavior of a Switched Poisson Arrival Queue (SPAQ) under overload control. The queue has infinite buffer capacity with an exponential server. A switched Poisson process is used to represent an aggregated voice/data packet arrival process in integrated services networks. By overload control, we mean to properly adapt the arrival process once the buffer contents exceed a designated level. The probability distribution of queue length as a function of time is obtained. The temporal effect of the overload control is measured in two forms. While in overload, we measure the amount of time for the queue to fall into an underload period. While in underload, we measure the amount of time for the queue to rise to an overload period. a proper design of the control will not only reduce the fall time but also increase the rise time, to protect high priority and time constrained services. Our study provides important information to overload control design on packet switching networks. We also explore the transient queueing behavior as affected by time stochastic properties of the underlying two-state Markov chain for the arrival process.

Transient analysis of multi-server queues with Markov-modulated Poisson arrivals and overload control

This paper studies the transient behavior of a Markov-molulated Poisson arrival queue under overload control. The queue has finite or infinite buffer capacity with multiple exponential servers. A Markov-modulated Poisson process is used to represent an aggregated voice or video packet arrival process in integrated service networks. By overload control, we mean to properly adapt the arrival process once the buffer contents exceed a designated level. The probability distribution of queue length as a function of time is obtained. The temporal effect of the overload control is measured in two forms. While in overload, we measure the amount of time for the queue to fall into underload. While in underload, we measure the amount of time for the queue to rise to overload. A proper design of the control will not only reduce the fall time but also increase the rise time. We also explore the transient queueing behavior as affected by time stochastic properties of the underlying Markov chain for the arrival process.

Queue length distributions in a markov model of a multistage clocked queueing network

AbstractIn [4], we treated the problem of passage through a discrete‐time clock‐regulated multistage queueing network by modeling the input time series {an} to each queue as a Markov chain. We showed how to transform probability transition information from the input of one queue to the input of the next in order to predict mean queue length at each stage. The Markov approximation is very good for p = E(an) ≦ ½, which is in fact the range of practical utility. Here we carry out a Markov time series input analysis to predict the stage to stage change in the probability distribution of queue length. The main reason for estimating the queue length distribution at each stage is to locate “hot spots”, loci where unrestricted queue length would exceed queue capacity, and a quite simple expression is obtained for this purpose.

Analytic and Numerical Solutions of an N Duplicate-Server System

We consider an N duplicate-server system, where each server consists of two reconfigurable duplicated units which are subject to breakdowns. This system is studied analytically using generating-functions, and also numerically using the matrix-geometric procedure. Using the generating-function approach we obtain a recursive expression of the queue-length probability distribution for N = 1. This expression is difficult to generalize to N ⩾ 2. The numerical method is applicable for any value of N. For any N, we also obtain the condition for stability and the availability of the system.