Non-stationary Poisson Arrivals Research Articles

Transforming Renewal Processes for Simulation of Nonstationary Arrival Processes

Simulation models of real-life systems often assume stationary (homogeneous) Poisson arrivals. Therefore, when nonstationary arrival processes are required, it is natural to assume Poisson arrivals with a time-varying arrival rate. For many systems, however, this provides an inaccurate representation of the arrival process that is either more or less variable than Poisson. In this paper we extend techniques that transform a stationary Poisson arrival process into a nonstationary Poisson arrival process (NSPP) by transforming a stationary renewal process into a nonstationary, non-Poisson (NSNP) arrival process. We show that the desired arrival rate is achieved and that when the renewal base process is either more or less variable than Poisson, then the NSNP process is also more or less variable, respectively, than an NSPP. We also propose techniques for specifying the renewal base process when presented properties of, or data from, an arrival process and illustrate them by modeling real arrival data.

A stationary Poisson departure process from a minimally delayed infinite server queue with non-stationary Poisson arrivals

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

A stationary Poisson departure process from a minimally delayed infinite server queue with non-stationary Poisson arrivals

The points of a non-stationary Poisson process with periodic intensity are independently shifted forward in time in such a way that the transformed process is stationary Poisson. The mean shift is shown to be minimal. The approach used is to consider an Mt/Gt/∞ queueing system where the arrival process is a non-stationary Poisson with periodic intensity function. A minimal service time distribution is constructed that yields a stationary Poisson departure process.

Transient laws of non-stationary queueing systems and their applications

In this paper we consider the general class of non-stationary queueing models and identify structural relationships between the number of customers in the system and the delay at time t, denoted by L(t) and S(t), respectively. In particular, we first establish a transient Little’s law at the same level of generality as the classical stationary version of Little’s law. We then obtain transient distributional laws for overtake free non-stationary systems. These laws relate the distributions of L(t) and S(t) and constitute a complete set of equations that describes the dynamics of overtake free non-stationary queueing systems. We further extend these laws to multiclass systems as well. Finally, to demonstrate the power of the transient laws we apply them to a variety queueing systems: Infinite and single server systems with non-stationary Poisson arrivals and general non-stationary services, multiclass single server systems with general non-stationary arrivals and services, and multiserver systems with renewal arrivals and deterministic services, operating in the transient domain. For all specific systems we relate the performance measures using the established set of laws and obtain a complete description of the system in the sense that we have a sufficient number of integral equations and unknowns. We then solve the set of integral equations using asymptotic expansions and exact numerical techniques. We also report computational results from our methods.

Stationary Poisson departure processes from non-stationary queues

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt/∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

Stationary Poisson departure processes from non-stationary queues

We present some non-stationary infinite-server queueing systems with stationary Poisson departure processes. In Foley (1982), it was shown that the departure process from the Mt/Gt/∞ queue was a Poisson process, possibly non-stationary. The Mt/Gt /∞ queue is an infinite-server queue with a stationary or non-stationary Poisson arrival process and a general server in which the service time of a customer may depend upon the customer's arrival time. Mirasol (1963) pointed out that the departure process from the M/G/∞ queue is a stationary Poisson process. The question arose whether there are any other Mt/Gt/∞ queueing systems with stationary Poisson departure processes. For example, if the arrival rate is periodic, is it possible to select the service-time distribution functions to fluctuate in order to compensate for the fluctuations of the arrival rate? In this situation and in more general situations, it is possible to select the server such that the system yields a stationary Poisson departure process.

On regenerative and ergodic properties of the k-server queue with non-stationary Poisson arrivals

We consider the stable k-server queue with non-stationary Poisson arrivals and i.i.d. service times and show that the non-time-homogeneous Markov process Zt = (the queue length and residual service times at time t) can be subordinated to a stable time-homogeneous regenerative process. As an application we show that if the system starts from given conditions at time s then the distribution of Zt stabilizes (but depends on t) as s tends backwards to –∞. Also moment and stochastic domination results are established for the delay and recurrence times of the regenerative process leading to results on uniform rates of convergence.

On regenerative and ergodic properties of the k-server queue with non-stationary Poisson arrivals

We consider the stable k-server queue with non-stationary Poisson arrivals and i.i.d. service times and show that the non-time-homogeneous Markov process Zt = (the queue length and residual service times at time t) can be subordinated to a stable time-homogeneous regenerative process. As an application we show that if the system starts from given conditions at time s then the distribution of Zt stabilizes (but depends on t) as s tends backwards to –∞. Also moment and stochastic domination results are established for the delay and recurrence times of the regenerative process leading to results on uniform rates of convergence.

On ross's conjectures about queues with non-stationary poisson arrivals

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

On ross's conjectures about queues with non-stationary poisson arrivals

Ross (1978) conjectured that the average delay in a single-server queue is larger when the arrival process is a non-stationary Poisson process than when it is a stationary Poisson process with the same rate. We present an example where equality obtains. When the number of waiting-positions is finite, Ross conjectured that the proportion of lost customers is greater in the nonstationary case. We present a counterexample to this conjecture.

Average delay in queues with non-stationary Poisson arrivals

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1. We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.

Average delay in queues with non-stationary Poisson arrivals

One of the major difficulties in attempting to apply known queueing theory results to real problems is that almost always these results assume a time-stationary Poisson arrival process, whereas in practice the actual process is almost invariably non-stationary. In this paper we consider single-server infinite-capacity queueing models in which the arrival process is a non-stationary process with an intensity function ∧(t), t ≧ 0, which is itself a random process. We suppose that the average value of the intensity function exists and is equal to some constant, call it λ, with probability 1.We make a conjecture to the effect that ‘the closer {∧(t), t ≧ 0} is to the stationary Poisson process with rate λ ' then the smaller is the average customer delay, and then we verify the conjecture in the special case where the arrival process is an interrupted Poisson process.