Spring 2018, Math 171

Week 10

Poisson Process Conditioning
1. Let $N(t)$ be a Poisson process with rate $\lambda$. Fix $0 \le n$, $s \le t$.
  1. Compute $\mathbb{P}(N(s)=m \mid N(t) = n)$ for $m \le n$. Give the name and parameters of the distribution.
    - (Answer) Binomial($n,\frac{s}{t}$)
  2. Compute $\mathbb{E}[N(s) \mid N(t) = n]$
    - (Answer) $n\frac{s}{t}$
2. Let $N(t)$ be a Poisson process with rate $\lambda$. Fix $0 \le m \le n$, $r \le s \le t$.
  1. Compute $\mathbb{P}(N(s)-N(r)=k \mid N(t) = n, N(r)=m)$. Give the name and parameters of the distribution.
    - (Answer) Binomial($n-m,\frac{s-r}{t-r}$)
  2. Compute $\mathbb{E}[N(s) \mid N(t) = n, N(r)=m]$
    - (Answer) $(n-m)\frac{s-r}{t-r}$
  3. Compute the expected amount if time between the first and last arrivals in $[r, t]$ given that $N(t) = n$ and $N(r)=m$.
    - (Solution) For convenience of writing, I will omit the condition on $N(t) = n$ and $N(r)=m$ in the expectations.
    - Let $L$ be the time of the last arrival in the interval, and $F$ be the time of the first arrival in the interval. Then we seek \[\begin{aligned}\mathbb{E}[L-F] &= \mathbb{E}[L] - \mathbb{E}[F]\\&=t - \mathbb{E}[t-L] - r - \mathbb{E}[F-r] \\&= (t-r) - \mathbb{E}[t-L] - \mathbb{E}[F-r].\end{aligned}\] By symmetry, we can see that $t-L$ (the time between the last arrival and the end of the interval) and $F-r$ (the time between the start of the interval and the first arrival) have the same distribution, so \[\mathbb{E}[L-F] = (t-r) - 2\mathbb{E}[F-r].\]
    - Since we have conditioned on $N(t) = n$ and $N(r)=m$, we know that there are $n-m$ arrivals uniformly distributed in the interval by the conditioning property of the poisson process. Therefore, letting $U_1, \dots U_{n-m}$ be i.i.d uniform$[0, t-r]$ random variables, we have \[\begin{aligned}\mathbb{E}[F-r] &= \mathbb{E}[\min(U_1, \dots U_{n-m})]\\&= \int_0^{t-r}P(\min(U_1, \dots U_{n-m}) > s) ds\\&=\int_0^{t-r}P(U_1>s, \dots U_{n-m}>s) ds\\&=\int_0^{t-r}P(U_1>s)^{n-m} ds\\&=\int_0^{t-r}\left(\frac{t-r-s}{t-r}\right)^{n-m} ds\\&=-(t-r)\frac{\left(\frac{t-r-s}{t-r}\right)^{n-m+1}}{n-m+1} \Big\vert_0^{t-r}\\&=\frac{t-r}{n-m+1}\end{aligned}\]
    - So finally, we have \[\begin{aligned}\mathbb{E}[L-F] &= (t-r) - 2\frac{t-r}{n-m+1}\\&=(t-r)\left(1-\frac{2}{n-m+1}\right)\end{aligned}\]
    - As a sanity check, we check that if $n-m=1$ we have $\mathbb{E}[L-F]=0$, and if $n-m=\infty$ we have $\mathbb{E}[L-F]=(t-r)$
Renewal Process
1. Cars queue at a gate. The lengths of the cars are i.i.d with distribution $F_L$ and mean $\mu$. Let $L \sim F_L$. Each successive car stops leaving a gap, distributed according to a uniform distribution on $(0, 1)$, to the car in front (or to the gate in the case of the car at the head of the queue). Consider the number of cars $N(t)$ lined up within distance t of the gate. Determine $\lim_{t \to \infty} \frac{\mathbb{E}[N(t)]}{t}$ if
  1. $L = c$ is a fixed constant
  2. $L$ is exponentially distributed with parameter $\lambda$
2. Potential customers arrive at a service kiosk in a bank as a Poisson process of rate $\lambda$. Being impatient, the customers leave immediately unless the assistant is free. Customers are served independently, with mean service time $\mu$.
  1. Find the mean time between the starts of two successive service periods.
    - (Answer) $\mu + 1/\lambda$
  2. Find the long run rate at which customers are served
    - (Answer) $\frac{1}{\mu + 1/\lambda}$
  3. What proportion of customers who arrive at the bank actually get served?
    - (Answer) $\frac{1}{\lambda\mu + 1}$
3. Customers arrive at a 24 hour restaurant (which has only one table) according to a poisson process with rate 1 party per hour. If the table is occupied, they leave immediately. If the table is unoccupied, they stay and eat. Customers spend an average of $20 each. The length of time a party stays is uniformly distributed in $[0, N/2 \mathrm{\ hours}]$ where $N$ is the number of people in the party. Answer the following questions for the cases when $N$ is uniform$\{2, 3, 4\}$ and when $N$ is geometric($1/2$).
  1. Let $\tau_k$ be the time between when the $(k-1)$st party finishes eating and the $(k)$th party finishes eating. Compute $\mathbb{E}[\tau_k]$
    - (Answer) $\mathbb{E}[N/4]+1$
  2. Let $X_k$ be the amount the $(k)$th party spends. Compute $\mathbb{E}[X_k]$.
    - (Answer) $20\mathbb{E}[N]$
  3. Let $S(t)$ the amount total amount paid by all parties by time $t$. Compute $\lim_{t\to \infty}\frac{S(t)}{t}$, and justify the validity of your computation.
    - (Answer) $\frac{20\mathbb{E}[N]}{\mathbb{E}[N/4]+1}$. Justify by SLLN
Useful Formulas
1. For a discrete random variable $X$ taking values in $\{0, 1, 2, \dots\}$, we have $\mathbb{E}[X] = \sum_{k=0}^\infty P(X > k)$
2. For a continuous random variable $T$ taking values in $[0, \infty)$, we have $\mathbb{E}[T] = \int_0^\infty P(T > s) ds$