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Occupancy Distribution as Seen by a Poisson Arrival

Module by: Bart Sinclair

Summary: (Blank Abstract)

The occupancy distribution of a system is the steady state probability distribution of the number of jobs in the system. We can also look at the system only at arrival instants (or to be precise, just before each arrival instant) and talk about the occupancy distribution of the system as seen by each arrival. When the arrivals are the results of a Poisson process and the system is open (the rate of arrivals is independent of the number of jobs in the system), we can show that the arrival time occupancy distribution is the same as the occupancy distribution over all time. In effect, there is nothing special about the arrival instants of a Poisson process.

This can be shown as follows:

Define Nt N t to be the number of jobs in an (open) system with Poisson arrivals at time tt.

Assume that the following limits exist. a n =limt→∞PrNt=n|arrival just after time t a n t Pr N t n | arrival just after time t p n =limt→∞PrNt=n p n t Pr N t n Then a n = p n a n p n .

Proof: Att+δ≡event that an arrival occurs in tt+δ A t t δ event that an arrival occurs in t t δ p n t≡PrNt=n p n t Pr N t n a n t≡PrNt=n|arrival just after t a n t Pr N t n | arrival just after t By the definition of conditional probability, a n t=limδ→0PrNt=n|Att+δ=limδ→0PrNt=n∧Att+δPrAtt+δ=limδ→0PrAtt+δ|Nt=nPrNt=nPrAtt+δ a n t δ 0 Pr N t n | A t t δ δ 0 Pr N t n A t t δ Pr A t t δ δ 0 Pr A t t δ | N t n Pr N t n Pr A t t δ For Poisson arrivals in an open system, we know that the number of arrivals in any interval tt+δ t t δ where δ>0 δ 0 is independent of Nt N t . ⇒ PrAtt+δ|Nt=n=PrAtt+δ Pr A t t δ | N t n Pr A t t δ ⇒ a n t=PrNt=n= p n t a n t Pr N t n p n t Hence, a n =limt→∞ a n t=limt→∞ p n t= p n a n t a n t t p n t p n

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