Queues and risk processes with dependencies

We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in [12]. In this setting, we obtain the steady state waiting time distribution and we show that the classical relation between the steady state waiting time and the workload distributions re- mains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally we show that there exist stochastic order relations between the waiting times under various instances of correlation

Diffusion Models for Double-ended Queues with Renewal Arrival Processes

We study a double-ended queue where buyers and sellers arrive to conduct trades. When there is a pair of buyer and seller in the system, they immediately transact a trade and leave. Thus there cannot be non-zero number of buyers and sellers simultaneously in the system. We assume that sellers and buyers arrive at the system according to independent renewal processes, and they would leave the system after independent exponential patience times. We establish fluid and diffusion approximations for the queue length process under a suitable asymptotic regime. The fluid limit is the solution of an ordinary differential equation, and the diffusion limit is a time-inhomogeneous asymmetric Ornstein-Uhlenbeck process (O-U process). A heavy traffic analysis is also developed, and the diffusion limit in the stronger heavy traffic regime is a time-homogeneous asymmetric O-U process. The limiting distributions of both diffusion limits are obtained. We also show the interchange of the heavy traffic and steady state limits

Many-server queues with customer abandonment: numerical analysis of their diffusion models

We use multidimensional diffusion processes to approximate the dynamics of a queue served by many parallel servers. The queue is served in the first-in-first-out (FIFO) order and the customers waiting in queue may abandon the system without service. Two diffusion models are proposed in this paper. They differ in how the patience time distribution is built into them. The first diffusion model uses the patience time density at zero and the second one uses the entire patience time distribution. To analyze these diffusion models, we develop a numerical algorithm for computing the stationary distribution of such a diffusion process. A crucial part of the algorithm is to choose an appropriate reference density. Using a conjecture on the tail behavior of a limit queue length process, we propose a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments also show that the diffusion models are good approximations for many-server queues, sometimes for queues with as few as twenty servers

Rare event analysis of Markov-modulated infinite-server queues: a Poisson limit

This article studies an infinite-server queue in a Markov environment, that is, an infinite-server queue with arrival rates and service times depending on the state of a Markovian background process. Scaling the arrival rates (i) by a factor N and the rates (ij) of the background process by N1+E (for some E>0), the focus is on the tail probabilities of the number of customers in the system, in the asymptotic regime that N tends to . In particular, it is shown that the logarithmic asymptotics correspond to those of a Poisson distribution with an appropriate mean

Application–Based Statistical Approach for Identifying Appropriate Queuing Model

Queuing theory is a mathematical study of queues or waiting lines. It is used to model many systems in different fields in our life, whether simple or complex systems. The key idea in queuing theory of a mathematical model is to improve performance and productivity of the applications. Queuing models are constructed in order to compute the performance measures for the applications and to predict the waiting times and queue lengths. This thesis is depended on previous papers of queuing theory for varies application which analyze the behavior of these applications and shows how to calculate the entire queuing statistic determined by measures of variability (mean, variance and coefficient of variance) for variety of queuing systems in order to define the appropriate queuing model. Computer simulation is an easy powerful tool to estimate approximately the proper queuing model and evaluate the performance measures for the applications. This thesis presents a new simulation model for defining the appropriate models for the applications and identifying the variables parameters that affect their performance measures. It depends on values of mean, variance and coefficient of the real applications, comparing them to the values for characteristics of the queuing model, then according to the comparison the appropriate queuing model is approximately identified.The simulation model will measure the effectiveness performance of queuing models A/B/1 where A is inter arrival distribution, B is the service time distributions of the type Exponential, Erlang, Deterministic and Hyper-exponential. The effectiveness performance of queuing model are: *L : The expected number of arrivals in the system. *Lq : The expected number of arrivals in the queue. *W : The expected time required a customer to spend in the system. *Wq : The expected time required a customer to spend in Queue. *U : the server utilization

Delays in IP routers, a Markov model

Delays in routers are an important component of end-to-end delay and therefore have a significant impact on quality of service. While the other component, the propagation time, is easy to predict as the distance divided by the speed of light inside the link, the queueing delays of packets inside routers depend on the current, usually dynamically changing congestion and on the stochastic features of the flows. We use a Markov model taking into account the distribution of the size of packets and self-similarity of incoming flows to investigate their impact on the queueing delays and their dynamics