Erlang Arrivals Research Articles

The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates

We study a queueing system with Erlang arrivals with k phases and Erlang service with m phases. Transition rates among phases vary periodically with time. For these systems, we derive an analytic solution for the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series. The method can be generalized to other level independent quasi-birth-death processes if the singularities of the generating function are known.

Erlang arrivals joining the shorter queue

We consider a system in which customers join upon arrival the shortest of two single-server queues. The interarrival times between customers are Erlang distributed and the service times of both servers are exponentially distributed. Under these assumptions, this system gives rise to a Markov chain on a multi-layered quarter plane. For this Markov chain we derive the equilibrium distribution using the compensation approach. The expression for the equilibrium distribution matches and refines tail asymptotics obtained earlier in the literature.

On the ruin time distribution for a Sparre Andersen process with exponential claim sizes

We derive a closed-form (infinite series) representation for the distribution of the ruin time for the Sparre Andersen model with exponentially distributed claims. This extends a recent result of Dickson et al. [Dickson, D.C.M., Hughes, B.D., Zhang, L., 2005. The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims. Scand. Actuar. J., 358–376] for such processes with Erlang inter-claim times. The derivation is based on transforming the original boundary crossing problem to an equivalent one on linear lower boundary crossing by a spectrally positive Lévy process. We illustrate our result in the cases of gamma, mixed exponential and inverse Gaussian inter-claim time distributions.

The density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claims

We derive expressions for the density of the time to ruin given that ruin occurs in a Sparre Andersen model in which individual claim amounts are exponentially distributed and inter-arrival times are distributed as Erlang(n, β). We provide numerical illustrations of finite time ruin probabilities, as well as illustrating features of the density functions.

An exploration of models that minimize leadtime through batching of arrived orders

In the last decade interest in work-in-process inventory control has grown. Many papers deal with this topic by considering the manufacturing leadtime as the critical factor that determines the amount of work-in-process. Several authors studied the influence of a batching decision on the average manufacturing leadtime. To this end queueing models with batch arrivals and batch service times were analyzed. One of the underlying assumptions made in the analysis is that the arrival process of the batches can be approximated by a Poisson process for each choice of the batchsize. However, when the interarrival times of individual clients are negative exponentially distributed an Erland distribution may be more appropriate as distribution of the interarrival time of the batches at the production unit. In this paper we consider the single item case. A very tractable analytical approximation for the average leadtime when batches arrive according to an Erlang distribution will be derived. Expressions for the optimal batchsize and the associated minimal leadtime are calculated and compared to experimental values obtained by simulation experiments. The approximation appears to be good. Finally, the huge differences in outcomes between Poisson and Erlang arrivals of the batches are highlighted.

Computer terminal memory performance in a resource sharing distributed supercomputing environment

Computer terminal memory performance in a resource sharing distributed supercomputing environment is analysed with Erlang arrivals, synchronous transmission and single server random interruptions through a Bernoulli sequence of independent random variables. A new method to derive a closed expression for data arrivals with Erlangly distributed interarrival times is presented. A simple and easy approach to derive the steady state probability expressions of the terminal memory content process is presented. The memory performance is studied in terms of overflow probability and average queueing delay. The results portrayed on graphs will provide an insight about system response time and performance to the system designers.

Modelling and analysis of display terminal at a finite user resource sharing node in distributed computer systems

An intelligent display terminal system at a finite user resource sharing node in distributed computer systems is modelled as a queueing model with finite microcomputer memory, burst Erlang arrivals from the display terminal cathode-ray-tube, constant service-time, synchronous transmission and single server interrupted through a first-order Markov process. A new generalized method differs from the usual stage and summing method used by the previous research-workers is presented to calculate burst Erlang arrival probabilities. The display terminal microcomputer memory performance is studied for finite and infinite cases and the expressions for overflow probabilities and average queueing delays are derived using the recursive method developed for calculating steady state probabilities of the microcomputer memory states. An integrated voice-data system is presented as a probable application of the model at remote areas and some of its computed values are portrayed on graphs to be useful in calculating response times of remote display terminals sharing voice channels for transmission purpose.

Simulation Studies for Data Multiplexing in Speech with Stdm or Tasi

A simulation model of a finite buffer queueing system with Erlang arrivals and multiple random servers is presented. The number of active servers in a service interval vary through a birth and death process derived from the speech samples in synchronous time division multiplexing (STDM) or time assignment speech interpolation (TASI) system. The results of this study show that the TASI system requires a larger buffer than the STDM system for a specified overflow probability.

Simulation model of buffer behavior for data multiplexing with Erlang arrivals in analog speech

Due to great simplicity in mathematical analysis, data-arrival process is generally assumed to be a Poisson process but it does not fit in many practical situations. However, the arrival process can be better approximated by Erlang distribution having appropriate degree. This paper analyses buffer behavior in queueing system with finite waiting-room, Erlang arrivals, synchronous transmission and single server having interruptions through a first order Markov process. Erlang distribution for arrival process puts multiple dependence on buffer states. Also this interruption process gives two simultaneous linear matrix equations. Thus the solution of state equations is difficult. Therefore, computer simulation has been used to study the relationships among overflow probability, buffer size and expected queueing delay due to buffering as a function of traffic intensity and the degree of Erlang distribution (Erlang parameter). A voice-data system (shown in Fig. 1) has been considered as an application for this model. The results of this study are portrayed on graphs and may be used as guidelines in buffer design problems in similar situations. Although this problem arose in the study of multiplexing of data in speech, the queueing model developed is quite general and may be useful for other industrial applications.

On buffer design for Erlang arrivals and multiple synchronous periodically regular outputs

The buffer design problem is studied for the case of Erlang arrivals, multiple, synchronous servicing channels, and periodically regular removal rates. Using traffic intensity, Erlang constant, and number of servicing channels as parameters the analytical expressions among buffer size, overflow probability, mean queue length and expected queuing delay due to buffering are derived. These relationships are computed for a sot of parameter values and are plotted to provide a basis for the buffer design. The results of the paper should be valuable in many engineering applications such as time-sharing computer systems, information compression, telephone exchanges, statistical multiplexing of data communication, and simulation of road traffic systems. As an example, a time-sharing computer system, servicing a large number of remote terminals, is studied.

Buffer length for Erlang arrivals and multiple synchronous regular removals

The problem of choosing the buffer length for a maximum allowed fractional loss, in a situation where the feeding rate is Erlangian, and where the service is effected through multiple output lines with regular and synchronous removal rates, is studied. Design curves are provided which can be used as a guide for the practical buffer-length choice.