Section 8.3

Exponential Variables and Races

Memorylessness, minima of exponentials, races between exponential clocks (the k-th arrival of one before the j-th of another).

Toolkit

Exponential clocks are memoryless: $\mathbb{P}(X>s+t\mid X>s)=\mathbb{P}(X>t)$ , so conditioning on survival past a threshold leaves a fresh exponential of the same rate. If $X_i\sim\mathrm{Exp}(\lambda_i)$ independent, then

\min_i X_i\sim\mathrm{Exp}\Big(\sum_i\lambda_i\Big),\qquad \mathbb{P}(X_j=\min_i X_i)=\frac{\lambda_j}{\sum_i\lambda_i},

and scaling $cX\sim\mathrm{Exp}(\lambda/c)$ . In a race between two independent Poisson streams of rates $\lambda,\mu$ , each arrival (from the pooled process) is independently “stream 1” with probability $\lambda/(\lambda+\mu)$ ; so the event “ $k$ -th arrival of stream 1 precedes $j$ -th arrival of stream 2” reduces to a negative-binomial/binomial comparison: stream 1 wins iff among the first $k+j-1$ pooled arrivals at least $k$ are of type 1, i.e.

\mathbb{P}(\text{stream 1 wins}) = \sum_{m=k}^{k+j-1}\binom{k+j-1}{m}p^m(1-p)^{k+j-1-m},\quad p=\frac{\lambda}{\lambda+\mu}.

Repeated independent races multiply probabilities.

Problems