The MM CPP/GE/c G-Queue: Sojourn Time Distribution

We obtain the sojourn time probability distribution function at equilibrium for a Markov modulated, multi-server, single queue with generalised exponential (GE) service time distribution and compound Poisson arrivals of both positive and negative customers. Such arrival processes can model both burstiness and correlated traffic and are well suited to models of ATM and other telecommunication networks. Negative customers remove (ordinary) customers in the queue and are similarly correlated and bursty. We consider both the cases where negative customers remove positive customers from the front and the end of the queue and, in the latter case, where a customer currently being served can and cannot be killed by a negative customer. These cases can model an unreliable server or load balancing respectively. The results are obtained as Laplace transforms and can be inverted numerically. The MM CPP/GE/c G-Queue therefore holds the promise of being a viable building block for the analysis of queues and queueing networks with bursty, correlated traffic, incorporating load balancing and node-failures, since the equilibrium behaviour of both queue lengths and response times can be determined in a tractable way.     

Product form solution for exponential G-networks with dependent service and completion of service of killed customers

Queueing networks with negative customers (G-networks), Poisson flow of positive customers, multi-server exponential nodes, and dependent service at the different nodes are studied. Every customer arriving at the network is defined by a set of random parameters: customer route, the length of customer route, customer volume and his service time at each route stage as well. A killed positive customer is removed at the last place in the queue and quits the network just after his remaining service time will be elaborated. For such G-networks, the multidimensional stationary distribution of the network state probabilities is shown to be representable in product form.     

Bibliography on G-networks,negative customers and applications

The idea of G-networks with negative arrivals, as well as of the relevant product form solution including non-linear traffic equations, was first published by Erol Gelenbe in 1989. In contrast to classical queues and queueing networks, the arrivals of negative customers which remove customers from a non-empty queue upon their arrival are possible in G-networks. Negative customers with appropriate killing discipline can be used to model breakdowns and to model packet losses, etc., while triggered customer movement can represent control processes in networks. This work presents a bibliography¹ on G-networks, negative customers and the use of G-networks, negative customers and triggers to various performance analysis problems. We hope that we can include a majority of publications on G-networks. This bibliography in the BibTex format and a grouping by various themes is available online from http://www.hit.bme.hu/~do/G-networks/. We would encourage readers and researchers to send information to the author in order to make this bibliography as complete as possible.     

On Kelly networks with shuffling

We consider Kelly networks with shuffling of customers within each queue. Specifically, each arrival, departure or movement of a customer from one queue to another triggers a shuffle of the other customers at each queue. The shuffle distribution may depend on the network state and on the customer that triggers the shuffle. We prove that the stationary distribution of the network state remains the same as without shuffling. In particular, Kelly networks with shuffling have the product form. Moreover, the insensitivity property is preserved for symmetric queues.      

A GI/Geo/1 queue with negative and positive customers

The arrival of a negative customer to a queueing system causes one positive customer to be removed if any is present. Continuous-time queues with negative and positive customers have been thoroughly investigated over the last two decades. On the other hand, a discrete-time Geo/Geo/1 queue with negative and positive customers appeared only recently in the literature. We extend this Geo/Geo/1 queue to a corresponding GI/Geo/1 queue. We present both the stationary queue length distribution and the sojourn time distribution.     

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.     

An Alternative Model for Queueing Systems with Single Arrivals,Batch Services and Customer Coalescence

In this paper we consider a queueing system with single arrivals, batch services and customer coalescence and we use it as a building block for constructing queueing networks that incorporate such characteristics. Chao et al. (1996) considered a similar model and they proved that it possesses a geometric product form stationary distribution, under the assumption that if the number of units present at a service completion epoch is less than the required number of units, then all the units coalesce into an incomplete (defective) batch which leaves the system. We drop this assumption and we study a model without incomplete batches. We prove that the stationary distribution of such a queue has a nearly geometric form. Using quasi-reversibility arguments we construct a network model with such queues which provides relevant bounds and approximations for the behaviour of assembly processes. Several issues about the validity of these bounds and approximations are also discussed.     

G-networks with multiple classes of signals and positive customers

G-networks are queueing models in which the types of customers one usually deals with in queues are enriched in several ways. In Gnetworks, positive customers are those that are ordinarily found in queueing systems; they queue up and wait for service, obtain service and then leave or go to some other queue. Negative customers have the specific function of destroying ordinary or positive customers. Finally triggers simply move an ordinary customer from one queue to the other. The term “signal” is used to cover negative customers and triggers. G-networks contain these three type of entities with certain restrictions; positive customers can move from one queue to another, and they can change into negative customers or into triggers when they leave a queue. On the other hand, signals (i.e. negative customers and triggers) do not queue up for service and simply disappear after having joined a queue and having destroyed or moved a negative customer. This paper considers this class of networks with multiple classes of positive customers and of signals. We show that with appropriate assumptions on service times, service disciplines, and triggering or destruction rules on the part of signals, these networks have a product form solution, extending earlier results.     

Queue Length Distribution in a FIFO Single-Server Queue with Multiple Arrival Streams Having Different Service Time Distributions

This paper considers the queue length distribution in a class of FIFO single-server queues with (possibly correlated) multiple arrival streams, where the service time distribution of customers may be different for different streams. It is widely recognized that the queue length distribution in a FIFO queue with multiple non-Poissonian arrival streams having different service time distributions is very hard to analyze, since we have to keep track of the complete order of customers in the queue to describe the queue length dynamics. In this paper, we provide an alternative way to solve the problem for a class of such queues, where arrival streams are governed by a finite-state Markov chain. We characterize the joint probability generating function of the stationary queue length distribution, by considering the joint distribution of the number of customers arriving from each stream during the stationary attained waiting time. Further we provide recursion formulas to compute the stationary joint queue length distribution and the stationary distribution representing from which stream each customer in the queue arrived.     

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.     

Prediction in Markovian bulk arrival queues

This paper deals with the statistical analysis of bulk arrival queues from a Bayesian point of view. The focus is on prediction of the usual measures of performance of the system in equilibrium. Posterior predictive distribution of the number of customers in the system is obtained through its probability generating function. Posterior distribution of the waiting time, in the queue and in the system, of the first customer of an arriving group is expressed in terms of their Laplace and Laplace–Stieltjes transform. Discussion of numerical inversion of these transforms is addressed.     

Two Queues with Weighted One-Way Overflow

We consider an open queueing network consisting of two queues with Poisson arrivals and exponential service times and having some overflow capability from the first to the second queue. Each queue is equipped with a finite number of servers and a waiting room with finite or infinite capacity. Arriving customers may be blocked at one of the queues depending on whether all servers and/or waiting positions are occupied. Blocked customers from the first queue can overflow to the second queue according to specific overflow routines. Using a separation method for the balance equations of the two-dimensional server and waiting room demand process, we reduce the dimension of the problem of solving these balance equations substantially. We extend the existing results in the literature in three directions. Firstly, we allow different service rates at the two queues. Secondly, the overflow stream is weighted with a parameter p ∈ [0,1], i.e., an arriving customer who is blocked and overflows, joins the overflow queue with probability p and leaves the system with probability 1 − p. Thirdly, we consider several new blocking and overflow routines. An erratum to this article can be found at     

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.     

Networks of infinite-server queues with nonstationary Poisson input

In this paper we focus on networks of infinite-server queues with nonhomogeneous Poisson arrival processes. We start by introducing a more general Poisson-arrival-location model (PALM) in which arrivals move independently through a general state space according to a location stochastic process after arriving according to a nonhomogeneous Poisson process. The usual open network of infinite-server queues, which is also known as a linear population process or a linear stochastic compartmental model, arises in the special case of a finite state space. The mathematical foundation is a Poisson-random-measure representation, which can be obtained by stochastic integration. It implies a time-dependent product-form result: For appropriate initial conditions, the queue lengths (numbers of customers in disjoint subsets of the state space) at any time are independent Poisson random variables. Even though there is no dependence among the queue lengths at each time, there is important dependence among the queue lengths at different times. We show that the joint distribution is multivariate Poisson, and calculate the covariances. A unified framework for constructing stochastic processes of interest is provided by stochastically integrating various functionals of the location process with respect to the Poisson arrival process. We use this approach to study the flows in the queueing network; e.g., we show that the aggregate arrival and departure processes at a given queue (to and from other queues as well as outside the network) are generalized Poisson processes (without necessarily having a rate or unit jumps) if and only if no customer can visit that queue more than once. We also characterize the aggregate arrival and departure processes when customers can visit the queues more frequently. In addition to obtaining structural results, we use the stochastic integrals to obtain explicit expressions for time-dependent means and covariances. We do this in two ways. First, we decompose the entire network into a superposition of independent networks with fixed deterministic routes. Second, we make Markov assumptions, initially for the evolution of the routes and finally for the entire location process. For Markov routing among the queues, the aggregate arrival rates are obtained as the solution to a system of input equations, which have a unique solution under appropriate qualifications, but not in general. Linear ordinary differential equations characterize the time-dependent means and covariances in the totally Markovian case.     

Equilibrium analysis in batch-arrival queues with complementary services

This paper studies a batch-arrival queue with two complementary services. The two services are complementary and any customer has no benefit from obtaining just one of them. To the best of the authors’ knowledge, there are no works contributed to the batch-arrival queues on analysis of the equilibrium behaviors in queueing systems by now. The properties of batch-arrival queues, which is more practical and universal in reality, induce different Nash equilibria under competition or monopoly compared with the single-arrival queues. We observe the joint effect of batch joining rate and cost structure on the behavior of customers and graphically interpret the equilibrium solutions under competition. Moreover, we discuss the model under three types of price structures and give comparisons from customer and server points.     

Economies of scale,competitiveness, and trade patterns within the European community: Clarendon Press,Oxford, 1983, 193 pages, £20.00

We consider two queues in series with input to each queue, which can be controlled by accepting or rejecting arriving customers. The objective is to maximize the discounted or average expected net benefit over a finite or infinite horizon, where net benefit is composed of (random) rewards for entering customers minus holding costs assessed against the customers at each queue. Provided that it costs more to hold a customer at the first queue than at the second, we show that an optimal policy is monotonic in the following senses: Adding a customer to either queue makes it less likely that we will accept a new customer into either queue; moreover moving a customer from the first queue to the second makes it more (less) likely that we will accept a new customer into the first (second) queue. Our model has policy implications for flow control in communication systems, industrial job shops, and traffic-flow systems. We comment on the relation between the control policies implied by our model and those proposed in the communicationa literature.     

The cyclic queue and the tandem queue

We consider a closed queueing network, consisting of two FCFS single server queues in series: a queue with general service times and a queue with exponential service times. A fixed number \(N\) of customers cycle through this network. We determine the joint sojourn time distribution of a tagged customer in, first, the general queue and, then, the exponential queue. Subsequently, we indicate how the approach toward this closed system also allows us to study the joint sojourn time distribution of a tagged customer in the equivalent open two-queue system, consisting of FCFS single server queues with general and exponential service times, respectively, in the case that the input process to the first queue is a Poisson process.     

Analysis of preemptive loss priority queues with preemption distance

In a queueing system with preemptive loss priority discipline, customers disappear from the system immediately when their service is preempted by the arrival of another customer with higher priority. Such a system can model a case in which old requests of low priority are not worthy of deferred service. This paper is concerned with preemptive loss priority queues in which customers of each priority class arrive in a Poisson process and have general service time distribution. The strict preemption in the existing model is extended by allowing the preemption distance parameterd such that arriving customers of only class 1 throughp — d can preempt the service of a customer of classp. We obtain closed-form expressions for the mean waiting time, sojourn time, and queue size from their distributions for each class, together with numerical examples. We also consider similar systems with server vacations.     

Equilibrium analysis of the observable queues with balking and delayed repairs

The equilibrium threshold balking strategies are investigated for the fully observable and partially observable single-server queues with server breakdowns and delayed repairs. Upon arriving, the customers decide whether to join or balk the queue based on observation of the queue length and status of the server, along with the consideration of waiting cost and the reward after finishing their service. By using Markov chain approach and system cost analysis, we obtain the stationary distribution of queue size of the queueing systems and provide algorithms in order to identify the equilibrium strategies for the fully and partially observable models. Finally, the equilibrium threshold balking strategies and the equilibrium social benefit for all customers are derived for the fully and partially observable system respectively, both with server breakdowns and delayed repairs.