A formal approach to queueing processes in the steady state and their applications

1979 ◽  
Vol 16 (2) ◽  
pp. 332-346 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Steady State ◽  
Point Processes ◽  
Formal Approach ◽  
Formal Treatment ◽  
Queueing Processes

Using the theory of point processes, we give a formal treatment to queueing processes in the steady state. Based on this result, we obtain invariance relations between several quantities in G/G/c queues. As applications, the finiteness of their moments is discussed for G/G′/c queues. The basic results and notations used in this paper are contained in the author's previous paper (Miyazawa (1977)).

A formal approach to queueing processes in the steady state and their applications

1979 ◽  
Vol 16 (02) ◽  
pp. 332-346 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Steady State ◽  
Point Processes ◽  
Formal Approach ◽  
Formal Treatment ◽  
Queueing Processes

Using the theory of point processes, we give a formal treatment to queueing processes in the steady state. Based on this result, we obtain invariance relations between several quantities in G/G/c queues. As applications, the finiteness of their moments is discussed for G/G′/c queues. The basic results and notations used in this paper are contained in the author's previous paper (Miyazawa (1977)).

On arrivals that see time averages: a martingale approach

1990 ◽  
Vol 27 (2) ◽  
pp. 376-384 ◽  
Author(s):  
Benjamin Melamed ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Point Process ◽  
Point Processes ◽  
Steady State Distribution ◽  
State Distribution ◽  
Martingale Theory ◽  
Stochastic Intensity ◽  
Sample Paths ◽  
Time Averages ◽  
Definition Of

This paper is a sequel to our previous paper investigating when arrivals see time averages (ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed process (assumed to be well defined) and basic properties of Riemann–Stieltjes integrals. Here similar results are obtained using the stochastic intensity associated with the martingale theory of point processes, as in Brémaud (1981). In the martingale framework, the ASTA result is almost an immediate consequence of the definition of a stochastic intensity. In a stationary framework, the results characterize the Palm distribution, but stationarity is not assumed here. Watanabe's (1964) martingale characterization of a Poisson process is also applied to establish a general version of anti–PASTA: if the points of the point process are appropriately generated by the observed process and the observed process is Markov with left-continuous sample paths, then ASTA implies that the point process must be Poisson.

Renewal processes, population dynamics, and unimodular trees

2019 ◽  
Vol 56 (2) ◽  
pp. 339-357
Author(s):  
François Baccelli ◽  
Antonio Sodre
Keyword(s):  
Population Dynamics ◽  
Steady State ◽  
Positive Integer ◽  
Finite Number ◽  
Random Graph ◽  
Infinite Number ◽  
Point Processes ◽  
Renewal Processes ◽  
Directed Tree ◽  
Random Structures

AbstractBased on a simple object, an i.i.d. sequence of positive integer-valued random variables {an}n∊ℤ, we introduce and study two random structures and their connections. First, a population dynamics, in which each individual is born at time n and dies at time n + an. This dynamics is that of a D/GI/∞ queue, with arrivals at integer times and service times given by {an}n∊ℤ. Second, the directed random graph Tf on ℤ generated by the random map f(n) = n + an. Assuming only that E [a0] < ∞ and P [a0 = 1] > 0, we show that, in steady state, the population dynamics is regenerative, with one individual alive at each regeneration epoch. We identify a unimodular structure in this dynamics. More precisely, Tf is a unimodular directed tree, in which f(n) is the parent of n. This tree has a unique bi-infinite path. Moreover, Tf splits the integers into two categories: ephemeral integers, with a finite number of descendants of all degrees, and successful integers, with an infinite number. Each regeneration epoch is a successful individual such that all integers less than it are its descendants of some order. Ephemeral, successful, and regeneration integers form stationary and mixing point processes on ℤ.

On arrivals that see time averages: a martingale approach

1990 ◽  
Vol 27 (02) ◽  
pp. 376-384 ◽  
Author(s):  
Benjamin Melamed ◽  
Ward Whitt
Keyword(s):  
Steady State ◽  
Point Process ◽  
Point Processes ◽  
Steady State Distribution ◽  
State Distribution ◽  
Martingale Theory ◽  
Stochastic Intensity ◽  
Sample Paths ◽  
Time Averages ◽  
Definition Of

This paper is a sequel to our previous paper investigating whenarrivals see time averages(ASTA) in a stochastic model; i.e., when the steady-state distribution of an embedded sequence, obtained by observing a continuous-time stochastic process just prior to the points (arrivals) of an associated point process, coincides with the steady-state distribution of the observed process. The relation between the two distributions was also characterized when ASTA does not hold. These results were obtained using the conditional intensity of the point process given the present state of the observed process (assumed to be well defined) and basic properties of Riemann–Stieltjes integrals. Here similar results are obtained using the stochastic intensity associated with the martingale theory of point processes, as in Brémaud (1981). In the martingale framework, the ASTA result is almost an immediate consequence of the definition of a stochastic intensity. In a stationary framework, the results characterize the Palm distribution, but stationarity is not assumed here. Watanabe's (1964) martingale characterization of a Poisson process is also applied to establish a general version of anti–PASTA: if the points of the point process are appropriately generated by the observed process and the observed process is Markov with left-continuous sample paths, then ASTA implies that the point process must be Poisson.

Point processes arising in vehicular traffic flow

1971 ◽  
Vol 8 (4) ◽  
pp. 809-814 ◽  
Author(s):  
Edward A. Brill
Keyword(s):  
Steady State ◽  
Stationary Point ◽  
Point Processes ◽  
Fixed Time ◽  
Traffic Model ◽  
Markov Renewal Process ◽  
Vehicular Traffic ◽  
State Point ◽  
Vehicular Traffic Flow ◽  
Stationary Point Processes

In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of a GI/G/s queue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the cases M/M/s (s = 1, 2, ···, ∞) and M/G/∞. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for the M/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady state M/G/∞ queue are independent at any fixed time; this is a result I have been unable to find in the literature.

Point processes arising in vehicular traffic flow

1971 ◽  
Vol 8 (04) ◽  
pp. 809-814 ◽  
Author(s):  
Edward A. Brill
Keyword(s):  
Steady State ◽  
Stationary Point ◽  
Point Processes ◽  
Fixed Time ◽  
Traffic Model ◽  
Markov Renewal Process ◽  
Vehicular Traffic ◽  
State Point ◽  
Vehicular Traffic Flow ◽  
Stationary Point Processes

In this paper we investigate the properties of stationary point processes motivated by the following traffic model. Suppose there is a dichotomy of slow and fast points (cars) on a road with limited overtaking. It is assumed that fast points are delayed behind (or are clustered at) a slow point in accordance with the principles of aGI/G/squeue, the order of service being irrelevant. Thus each slow point represents a service station, with the input into each station consisting of a fixed (but random) displacement of the output of the previous queueing station. It is found that tractable results for stationary point processes occur for the casesM/M/s(s= 1, 2, ···, ∞) andM/G/∞. In particular, it is found that for these cases the steady state point processes are compound Poisson and that for theM/M/1 case the successive headways form a two state Markov renewal process. In addition it is shown that the input, output, and queue size processes in a steady stateM/G/∞queue are independent at any fixed time; this is a result I have been unable to find in the literature.

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