Stochastic Arrival Process
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A Stochastic Arrival Process is a stochastic decision process for a queuing system.
- Context:
- It can (typically) involve an Arrival Rate.
- Example(s):
- See: Stochastic Decision Process, Stochastic Process.
References
2011
- (MIT EECS, 262) ⇒ MIT (2011). “Chapter 2 - Poisson Process."
- QUOTE: An arrival process is a sequence of increasing rv’s, [math]\displaystyle{ 0 \lt S1 \lt S2 \lt }[/math], where[1] [math]\displaystyle{ S_i \lt S_{i+1} }[/math] means that Si+1-Si is a positive rv, i.e., a rv X such that FX(0) = 0. The rv’s S1,S2,... , are called arrival epochs (the word time is somewhat overused in this subject) and represent the times at which some repeating phenomenon occurs. Note that the process starts at time 0 and that multiple arrivals can’t occur simultaneously (the phenomenon of bulk arrivals can be handled by the simple extension of associating a positive integer rv to each arrival). We will sometimes permit simultaneous arrivals or arrivals at time 0 as events of zero probability, but these can be ignored. In order to fully specify the process by the sequence [math]\displaystyle{ S_1, S_2, ... }[/math] of rv’s, it is necessary to specify the joint distribution of the subsequences S1,... ,Sn for all n> 1.
Although we refer to these processes as arrival processes, they could equally well model departures from a system, or any other sequence of incidents. Although it is quite common, especially in the simulation field, to refer to incidents or arrivals as events, we shall avoid that here. The nth arrival epoch Sn is a rv and {Sn ? t}, for example, is an event. This would make it confusing to refer to the nth arrival itself as an event.
- QUOTE: An arrival process is a sequence of increasing rv’s, [math]\displaystyle{ 0 \lt S1 \lt S2 \lt }[/math], where[1] [math]\displaystyle{ S_i \lt S_{i+1} }[/math] means that Si+1-Si is a positive rv, i.e., a rv X such that FX(0) = 0. The rv’s S1,S2,... , are called arrival epochs (the word time is somewhat overused in this subject) and represent the times at which some repeating phenomenon occurs. Note that the process starts at time 0 and that multiple arrivals can’t occur simultaneously (the phenomenon of bulk arrivals can be handled by the simple extension of associating a positive integer rv to each arrival). We will sometimes permit simultaneous arrivals or arrivals at time 0 as events of zero probability, but these can be ignored. In order to fully specify the process by the sequence [math]\displaystyle{ S_1, S_2, ... }[/math] of rv’s, it is necessary to specify the joint distribution of the subsequences S1,... ,Sn for all n> 1.
- ↑ 1These rv’s Si can be viewed as sums of interarrival times. They should not be confused with the rv’s Si used in Section 1.3.5 to denote the number of arrivals by time i for the Bernoulli process. We use Si throughout to denote the sum of i rv’s. Understanding how such sums behave is a central issue of every chapter (and almost every section ) of these notes. Unfortunately, for the Bernoulli case, the IID sums of primary interest are the sums of binary rv’s at each time increment, whereas here the sums of primary interest are the sums of interarrival intervals.