Poisson Process

Counting Process

  1. $N(t)\geq 0$
  2. $N(t)\in\mathbb{N}$
  3. $s<t,N(s)\leq N(t)$
  4. $s<t,N(t)-N(s)$ equals the number of events that have occurred in the interval $(s,t]$

Definition 1 of Poisson Process

  1. $N(0)=0$

  2. independent increments

  3. $\forall s,t\geq 0$

    $$P(N(t+s)-N(s)=n)=e^{-\lambda t}\frac{(\lambda t)^n}{n!},n=0,1$$

Definition 2 of Poisson Process

  1. $N(0)=0$
  2. The process has stationary and independent increments
  3. $P(N(t+h)-N(t)=1)=\lambda h+o(h)$
  4. $P(N(t+h)-N(t)\geq 2)=o(h)$

Sequence of interarrival times

  • ${X_n,n\geq 1}$: $X_n$ denotes the time of the $n$th event

    $$P(X_1>t)=P(N(t)=0)=e^{-\lambda t}$$

  • $X_n$ are independent identically distributed exponential random variables having mean $\frac{1}{\lambda}$

Arrival time of $n$th event

  • $S_n=\sum_{i=1}^nX_i,n\geq 1$: Gamma Distribution

  • order statistics $Y_{(i)}$ corresponding to $Y_i$: $Y_{(i)}$ is the $k$th minimum of $Y_{i}$

    $$ f(y_1,y_2,\cdots,y_n)=n!\prod_{i=1}^nf(y_i) $$

  • $Y_i$ are uniformly distributed over $(0,t)$, then the joint density funciton of order statistics is

    $$ f(y_1,y_2,\cdots,y_n)=\frac{n!}{t^n} $$

  • Given that $N(t)=n$, the $n$ arrival times $S_1,\cdots,S_n$ have the same distribution as the order statistics corresponding to $n$ independent random variables uniformly distributed on the interval $(0,t)$

Two Type Poisson random variables

  • event occurs at time $s$, then, indepentently of all else, it is classified as being a type-I event with probability $P(s)$ and a type-II event with probability $1-P(s)$
  • $N_1(t)$ and $N_2(t)$ are independent Poisson random variables having respective means $\lambda tp$ and $\lambda t(1-p)$

$$ p=\frac{1}{t}\int^t_0P(s)ds $$

M/G/1 Busy Period

  • M/G/1 queueing system
    • customers arrive in accordance with a Poisson process with rate $\lambda$
    • upon arrival, either enter service if the server is free or else join the queue
    • successive service times are independent and identically distributed according to $G$: $Y_i$ denotes the sequence of service times
    • busy period begins: an arrival finds the server free
    • busy period ends: no longer any customers in the system
  • busy period last a time $t$ and consits of $n$ services (Probability $B(t,n)$) iff
    • $S_k\leq Y_1+\cdots Y_k$
    • $Y_1+\cdots Y_n=t$
    • There are $n-1$ arrivals in $(0,t)$

$$B(t,n)=\int_0^te^{-\lambda t}\frac{(\lambda t)^{n-1}}{n!}dG_n(t)$$

$$B(t)=\sum_{n=1}^\infty B(t,n)$$