# Spring 2018, Math 171

#### Week 10

1. 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 $$\mathbb{E}[L-F]=(t-r)$$
2. 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
3. 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 $$\mathbb{E}[T] = \int_0^\infty P(T > s) ds$$