400 research outputs found
Rare-event analysis of mixed Poisson random variables, and applications in staffing
A common assumption when modeling queuing systems is that arrivals behave
like a Poisson process with constant parameter. In practice, however, call
arrivals are often observed to be significantly overdispersed. This motivates
that in this paper we consider a mixed Poisson arrival process with arrival
rates that are resampled every time units, where and a
scaling parameter. In the first part of the paper we analyse the asymptotic
tail distribution of this doubly stochastic arrival process. That is, for large
and i.i.d. arrival rates , we focus on the evaluation of
, the probability that the scaled number of arrivals exceeds .
Relying on elementary techniques, we derive the exact asymptotics of :
For we identify (in closed-form) a function
such that tends to as .
For and we find a partial
solution in terms of an asymptotic lower bound. For the special case that the
s are gamma distributed, we establish the exact asymptotics across all . In addition, we set up an asymptotically efficient importance sampling
procedure that produces reliable estimates at low computational cost. The
second part of the paper considers an infinite-server queue assumed to be fed
by such a mixed Poisson arrival process. Applying a scaling similar to the one
in the definition of , we focus on the asymptotics of the probability
that the number of clients in the system exceeds . The resulting
approximations can be useful in the context of staffing. Our numerical
experiments show that, astoundingly, the required staffing level can actually
decrease when service times are more variable
Combined analysis of transient delay characteristics and delay autocorrelation function in the Geo(X)/G/1 queue
We perform a discrete-time analysis of customer delay in a buffer with batch arrivals. The delay of the kth customer that enters the FIFO buffer is characterized under the assumption that the numbers of arrivals per slot are independent and identically distributed. By using supplementary variables and generating functions, z-transforms of the transient delays are calculated. Numerical inversion of these transforms lead to results for the moments of the delay of the kth customer. For computational reasons k cannot be too large. Therefore, these numerical inversion results are complemented by explicit analytic expressions for the asymptotics for large k. We further show how the results allow us to characterize jitter-related variables, such as the autocorrelation of the delay in steady state
Universality for mathematical and physical systems
All physical systems in equilibrium obey the laws of thermodynamics. In other
words, whatever the precise nature of the interaction between the atoms and
molecules at the microscopic level, at the macroscopic level, physical systems
exhibit universal behavior in the sense that they are all governed by the same
laws and formulae of thermodynamics. In this paper we describe some recent
history of universality ideas in physics starting with Wigner's model for the
scattering of neutrons off large nuclei and show how these ideas have led
mathematicians to investigate universal behavior for a variety of mathematical
systems. This is true not only for systems which have a physical origin, but
also for systems which arise in a purely mathematical context such as the
Riemann hypothesis, and a version of the card game solitaire called patience
sorting.Comment: New version contains some additional explication of the problems
considered in the text and additional reference
Large Deviations and Transient Multiplexing at a Buffered Resource
In this paper we discuss asymptotics associated with a large number of sources using a resource in a compact time interval. A large deviations condition is placed on the sum of the vectors that describe the stochastic behaviour of the sources and large deviations results deduced about the probability of exhaustion of the resource. This approach allows us to consider sources which are highly non-stationary in time. The examples in mind are a single server queue and a form of the Cramer-Lundburg model from risk theory. Connection is made with past work on stability of queues and effective bandwidths. A number of examples are presented to illustrate the strengths of this approach
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