Poisson Arrivals See Time Averages Research Articles

A two-stage queueing network with MAP inputs and buffer sharing

In this paper, we study a two-stage tandem queueing network with MAP inputs and buffer sharing. The two stages share the same buffer. By using Markov process, we give an exact analysis of the queueing network. Since the customer arrival is not a Poisson process, the PASTA (Poisson Arrivals See Time Averages) property does not hold. A matrix filtration technique is proposed to derive the probability distribution of queue length at arrivals. Our objective is to investigate how the buffer sharing policy is mitigate the tradeoff between the probability that an arriving customer is lost and the probability that the first-stage server is blocked. The numerical results show that buffer sharing policy is more flexible, especially when the inputs have large variant and are correlated.

The Role of PASTA in Network Measurement

Poisson arrivals see time averages (PASTA) is a well-known property applicable to many stochastic systems. In active probing, PASTA is invoked to justify the sending of probe packets (or trains) at Poisson times in a variety of contexts. However, due to the diversity of aims and analysis techniques used in active probing, the benefits of Poisson-based measurement, and the utility and role of PASTA, are unclear. Using a combination of rigorous results and carefully constructed examples and counterexamples, we map out the issues involved and argue that PASTA is of very limited use in active probing. In particular, Poisson probes are not unique in their ability to sample without bias. Furthermore, PASTA ignores the issue of estimation variance and the central need for an inversion phase to estimate the quantity of interest based on what is directly observable. We give concrete examples of when Poisson probes should not be used, explain why, and offer initial guidelines on suitable alternative sending processes.

The role of PASTA in network measurement

Poisson Arrivals See Time Averages (PASTA) is a well known property applicable to many stochastic systems. In active probing, PASTA is invoked to justify the sending of probe packets (or trains) at Poisson times in a variety of contexts. However, due to the diversity of aims and analysis techniques used in active probing, the benefits of Poisson based measurement, and the utility and role of PASTA, are unclear. Using a combination of rigorous results and carefully constructed examples and counter-examples, we map out the issues involved, and argue that PASTA is of very limited use in active probing. In particular, Poisson probes are not unique in their ability to sample without bias. Furthermore, PASTA ignores the issue of estimation variance, and the central need for an inversion phase to estimate the quantity of interest ased on what is directly observable. We give concrete examples of when Poisson probes should not be used, and explain why, and offer initial guidelines on suitable alternative sending processes.

How to couple from the past using a read-once source of randomness

We give a new method for generating perfectly random samples from the stationary distribution of a Markov chain. The method is related to coupling from the past (CFTP), but only runs the Markov chain forwards in time, and never restarts it at previous times in the past. The method is also related to an idea known as PASTA (Poisson arrivals see time averages) in the operations research literature. Because the new algorithm can be run using a read-once stream of randomness, we call it read-once CFTP. The memory and time requirements of read-once CFTP are on par with the requirements of the usual form of CFTP, and for a variety of applications the requirements may be noticeably less. Some perfect sampling algorithms for point processes are based on an extension of CFTP known as coupling into and from the past; for completeness, we give a read-once version of coupling into and from the past, but it remains unpractical. For these point process applications, we give an alternative coupling method with which read-once CFTP may be efficiently used. ©2000 John Wiley & Sons, Inc. Random Struct. Alg., 16: 85–113, 2000

Estimating Customer and Time Averages

In this paper we establish a joint central limit theorem for customer and time averages by applying a martingale central limit theorem in a Markov framework. The limiting values of the two averages appear in the translation terms. This central limit theorem helps to construct confidence intervals for estimators and perform statistical tests. It thus helps determine which finite average is a more asymptotically efficient estimator of its limit. As a basis for testing for PASTA (Poisson arrivals see time averages), we determine the variance constant associated with the central limit theorem for the difference between the two averages when PASTA holds.

A martingale approach to the PASTA property

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L 2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

A martingale approach to the PASTA property

It is known (Weizsäcker and Winkler (1990)) that for bounded predictable functions H and a Poisson process with jump times exists almost surely, and that in this case both limits are equal. Here we relax the boundedness condition on H. Our tool is a law of large numbers for local L2-martingales. We show by examples that our condition is close to optimal. Furthermore we indicate a generalization to point processes on more general spaces. The above property is called PASTA (‘Poisson arrivals see time averages') and is heavily used in queueing theory.

A filtered ASTA property

Recently, the PASTA (Poisson Arrivals See Time Averages) property has been extended to ASTA (Arrivals See Time Averages) by eliminating the need for Poisson arrivals and weakening the LAA (Lack of Anticipation Assumption). This paper presents a strengthening of ASTA under the original LAA of Wolff. We consider a stochastic processX with an associated point processN that admits a stochastic intensity and satisfies LAA. Various authors have noted in various contexts that ASTA holds if and only if the arrival intensity is state independent. For a class of point processes that includes doubly stochastic as well as ordinary Poisson processes, we prove that the point process obtained by restricting the processX to any given set of states not only has the same intensity but also the same probabilistic structure as the original point process. In particular, if the original point process is Poisson, the new point process is still Poisson with the same parameter as the original point process. For a discrete-time version, of interest in its own right, we provide a simple proof of a strengthened version of ASTA in discrete time. Unlike other discrete-time versions of ASTA, ours is valid for point processes with stationary but not necessarily independent increments. The continuous-time results are obtained using martingale theory. A corollary is a simple proof of PASTA under conditions that require only that the relevant limits exist. Our results may also provide some insight into characterizing Poisson flows in queueing systems.

On Wolff's Pasta Martingale

In establishing the seminal result PASTA (Poisson Arrivals See Time Averages), R. Wolff (1982) constructed a martingale, and demonstrated that PASTA was a consequence of a strong law of large numbers of this martingale. Here we establish a central limit theorem for the PASTA martingale, and characterize its asymptotic normality. The result can be used to construct confidence intervals for estimators of the difference between the arrival (event) and time averages.

Stationary Stochastic Models.

Recursive stochastic equations - weak and strong solutions - ergodic weak solutions - construction of stationary weak and strong solutions - Wald's identity for dependent random variables - model continuity in the presence of renewing epochs - method of metric modification - continuity of weak solutions the case of stationary Markov chains stationary sequences and stationary marked point processes - ergodic marked point processes continuous time models - stochastic processes with an embedded point process - semi-Markov and semi-regenerative processes - continuous time state processes - recursive stochastic equations in continuous time arrival-stationary queuing processes: existence and uniqueness - notations - the system - single server queue - the system with FCFS queueing discipline - the system with cyclic queueing discipline - the single server queue with warming-up - the many server loss system with repeated call attempts - open networks of loss systems relationships between arrival, time and departure stationary queueing processes - Poisson arrivals see time averages (PASTA) - the number of customers - the busy cycle - Takaes' and Pollaczek-Khinchin formulae batch-arrival stationary queuing processes - the system with geometrically distributed batch size - constant batch size - single server queue with batch arrivals - feedback queues - comparing batch delays and customer delays continuity of queueing models further models - the stochastic equation with stationary coefficients - stability of robust filter cleaners - non-contractivity of the filters - a further method for constructing solutions of recursive stochastic equations - the filter with fixed scale - variable scale.

Stationary Stochastic Models

"Stationary Stochastic Models." Journal of the Operational Research Society, 42(10), pp. 905–906

A Note on PASTA and Anti-PASTA for Continuous-Time Markov Chains

The result Poisson Arrivals See Time Averages (PASTA) is well established in the queueing literature. Although there are some well known examples where non-Poisson arrivals see time averages, arrival and time averages usually are different when arrivals are not Poisson. There are also situations where it has been shown that arrivals see time averages only if arrivals are Poisson, a property called Anti-PASTA. We present a simple and direct proof of Anti-PASTA for continuous-time Markov chains.

An Anti-PASTA Result for Markovian Systems

PASTA (Poisson Arrivals See Time Averages) is a term coined by R. Wolff in his well known 1982 paper. In keeping with Wolff's terminology, we use the term anti-PASTA to refer to the following converse of PASTA. Given that arrivals do indeed see time averages, when must the arrival process necessarily be Poisson? We show that anti-PASTA is satisfied in a pure-jump Markov process, provided that the arrival process corresponds to a subset of the Markov process jumps.

On the “pasta” property and a further relationship between customer and time averages in stationary queueing systems

We consider queueing systems in statistical equilibrium and discuss a property which is known as PASTA (Poisson Arrivals See Time Averages). We give examples showing that for this property as well as for a related property called Conditional PASTA to be valid the independence assumption usually made in this connection is not necessary. Furthermore, we extend to queueing systems with preemptive service discipline a formula of Brumelle's relating the time-average workload to the delay in queue and the service time of a customer.

Batch Delays Versus Customer Delays

For a large class of contention schemes with messages transmitted by subdividing them into packets, we show that the delay distribution of a message is the same as that for an individual packet. We conclude this from analyzing queueing systems with batch arrivals, where batch sizes have a geometric distribution and the queue discipline is indifferent to batch sizes and service times. There we prove that the customer (packet) delay distribution is the same as the batch (message) delay distribution, where delay is defined to be the delay of the last served customer in the batch. The proof is based on the discrete-time analog of the Poisson Arrivals See Time Averages (PASTA) theorem. We conclude that, in many cases, we can obtain message delays by calculating or measuring the packet delays, which is usually an easier task.