Arbitrary Point Process Research Articles

Managing the Callback Option under Arrival Rate Uncertainty

This paper studies the callback option as an instrument to effectively mitigate congestion due to temporary surges in arrivals to a call center. The call arrival process can be an arbitrary point process, allowing uncertainty and temporary surges in the arrival rate, provided that the system is stable. When a customer arrives, the call center manager examines the system state and decides whether to offer the incoming customer the callback option or not. When the callback option is offered, the customer decides whether to accept the offer. The customer is routed to the offline queue (to be called back later) only if he is offered the callback option and accepts it. Otherwise, he is routed to the online queue. For each customer in the online queue, the call center manager incurs a waiting cost of h per time unit. Similarly, whenever she routes a customer to the offline queue (for a callback later), she incurs a one-time penalty of p. Initially, we allow complete foresight policies that look into the entire future. We show that a simple lookahead policy that looks into the future arrivals and service completion times for the next p/h time units and uses the current number of customers in the system who previously rejected a callback offer (but does not look into the accept/reject decisions of future customers) is pathwise optimal. Building on the insights gleaned from the optimal lookahead policies, we also propose a non-anticipating (and implementable) policy by interpreting the lookahead policy in the fluid model and show its optimality in the fluid model. In particular, we show that this policy reduces to the so called line policy, if the arrival rate process follows a Cox-Ingersoll-Ross process. Lastly, we conduct a simulation study which shows that the proposed policies perform well.

Non-parametric optimal service pricing: a simulation study

In this paper, we study a price discovery algorithm for searching the optimal price for a service with price-sensitive demand. The customer’s response to price is unknown and the customer arrival process follows an arbitrary point process. This algorithm is suitable for new services with no prior knowledge or historical data available. Furthermore, there is no information about objective function as well. We take on a simulation study and discuss the sensitivity and robustness of this procedure with respect to different arrival processes and customer response functions and also provide the comparative statics with simple price learning algorithm. We prove that price discovery algorithm is a better convergent as it reduces stochastic error at each step. The main focus of this research is to provide some guidance for the selection of sample sizes based on the test significance and the measure of its power when actual mean and variance for revenue are unknown.

A note on lateral transshipment under point-process driven demand

We analyze lateral transshipment for items whose demand is driven by an arbitrary point process. We find conditions under which lateral transshipment stochastically lowers the cost of a system of demand centers. Our main structural result is that transshipment works best for slow-moving items, such as spare-parts, when the demands are driven by non-homogeneous or homogeneous Poisson processes. We also obtain general results that apply when demand is driven by an arbitrary point process.

Coding and Decoding with Adapting Neurons: A Population Approach to the Peri-Stimulus Time Histogram

The response of a neuron to a time-dependent stimulus, as measured in a Peri-Stimulus-Time-Histogram (PSTH), exhibits an intricate temporal structure that reflects potential temporal coding principles. Here we analyze the encoding and decoding of PSTHs for spiking neurons with arbitrary refractoriness and adaptation. As a modeling framework, we use the spike response model, also known as the generalized linear neuron model. Because of refractoriness, the effect of the most recent spike on the spiking probability a few milliseconds later is very strong. The influence of the last spike needs therefore to be described with high precision, while the rest of the neuronal spiking history merely introduces an average self-inhibition or adaptation that depends on the expected number of past spikes but not on the exact spike timings. Based on these insights, we derive a ‘quasi-renewal equation’ which is shown to yield an excellent description of the firing rate of adapting neurons. We explore the domain of validity of the quasi-renewal equation and compare it with other rate equations for populations of spiking neurons. The problem of decoding the stimulus from the population response (or PSTH) is addressed analogously. We find that for small levels of activity and weak adaptation, a simple accumulator of the past activity is sufficient to decode the original input, but when refractory effects become large decoding becomes a non-linear function of the past activity. The results presented here can be applied to the mean-field analysis of coupled neuron networks, but also to arbitrary point processes with negative self-interaction.

Stability of a Class of Hybrid Linear Stochastic Systems

The objective of the paper is to establish L q-stability results for the state processes of a new class of slowly time-varying hybrid continuous-time linear stochastic systems. The systems are hybrid in the sense that jumps both in the dynamics and the state may occur. In both cases a jump amounts to resetting to an initial value. Jump-times are defined by a more or less arbitrary point process. Most of the analysis is based on a carefully organized stochastic Lyapunov-function argument. We motivate our study by a brief summary of some basic problems in continuous-time recursive estimation of linear stochastic systems within a framework analogous to the one developed by Benveniste et al.

Maximum likelihood detection of PPM signals governed by arbitrary point-process plus additive Gaussian noise

The maximum likelihood decision statistic for pulse-position modulated (PPM) signals governed by an arbitrary discrete point process in the presence of additive Gaussian noise is derived. Sufficient conditions are given for determining when the optimum PPM symbol detection strategy is to choose the PPM symbol corresponding to the maximum slot statistic. In particular, it is shown that for the important case of Webb distributed avalanche photodiode output electrons in the presence of Gaussian noise, the optimum decision rule is to choose the largest slot observable.

Thinning of Point Processes—Martingale Method

By using the martingale method, we show that thinning of an arbitrary point process produces independent thinned processes if and only if the original point process is (nonhomogeneous) Poisson. Thinning is a classical problem for point processes. It is well-known that independent homogeneous Poisson processes result from constant Bernoulli thinnings of homogeneous Poisson processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. processes. In fact, the conclusion remains true for nonconstant Bernoulli thinnings of nonhomogeneous Poisson processes. It is easy to see that the converse is also true, i.e., if the thinned processes are independent nonhomogeneous Poisson processes, so are the original processes. But if we only suppose the thinned processes are independent, nothing is concerned with their distribution law, the problem of whether or not the original processes are (nonhomogeneous) Poisson becomes interesting and challenging, which is the objective of this paper. It is considerably surprising for us to arrive at the affirmative answer. So far as this problem is concerned, the most work was done under the renewal assumption. For example, Bremaud [1] showed that for arbitrary delayed renewal processes, the existence of a pair of uncorrelated thinned processes is sufficient to guarantee that the original process is Poisson. It is natural that the mathematical tools to solve the problem in this case be typical ones for renewal theory, such as renewal equations and Laplace-Stieltjes transformations. Obviously, they are not available for nonrenewal processes. We find out that the martingale method is the most efficient one to solve this problem in the general case. More precisely, we use mainly the dual predictable projections of point processes. In fact, the distribution law of a point process is determined uniquely by its dual predictable projection (see [5]), and Ma [6] offered us a very useful criterion of independence of jump processes having no common jump time through their dual predictable projections. Based on these results, it is not a long way to arrive at the destination.

Comparing ordered-entry queues with heterogeneous servers

We study a queueing system withm exponential servers with distinct service rates. Jobs arrive at the system following an arbitrary point process. Arrived jobs receive service at the first unoccupied server (if any) according to an entry order π, which is a permutation of the integers 1, 2,...,m. The system has a finite buffer capacity. When the buffer limit is reached, arrivals will be blocked. Blocked jobs will either be lost or come back as New arrivals after a random travel time. We are concerned with the dynamic stochastic behavior of the system under different entry orders. A partial ordering is established among entry orders, and is shown to result in some quite strong orderings among the associated stochastic processes that reflect the congestion and the service characteristics of the system. The results developed here complement existing comparison results for queues with homogeneous servers, and can be applied to aid the design of conveyor and communication systems.

Thinning of point processes—covariance analyses

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

Thinning of point processes—covariance analyses

Cross-covariances between the Bernoulli thinned processes of an arbitrary point process are determined. When the point process is renewal it is shown that zero correlation implies independence. An example is given to show that zero covariance between intervals does not imply zero covariance between counts. Mark-dependent thinning of Markov renewal processes is discussed and the results are applied to the overflow queue. Here we give an example of two uncorrelated but dependent renewal processes, neither of which is Poisson, which yield a Poisson process when superposed. Finally, we study Markov-chain thinning of renewal processes.

Insensitive Generalized Semi-Markov Schemes with Point Process Input

Where sequences of i.i.d. random variables are used for modelling successive repair limes, life times, service times, etc, it can happen that stationary probabilities are found for the—suitably defined—system state, which depend only on the means of those variables, not on any further characteristics of their distributions. The well-known Erlang loss model provides an example. For a large class of models, Jansen, Koenig, and Nawrotzki (Jansen, U., D. Koenig, K. Nawrotzki. 1979. A criterion of insensitivity for a class of queuing systems with random marked point processes. Math. Operationsforsch. Statist. 10 379–403.) have shown that, whenever such insensitivity prevails towards an i.i.d. sequence or—equivalently—a renewal process, then replacing this process by an arbitrary point process with the same intensity (when stationary) still does not change the stationary distribution of the system state. We extend their model and prove the analogous result by way of a different technique of general interest.