In Example 5.3 if server i serves at an exponential rate λi, i=1,2, show that P{ Smith is not la

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In Example 5.3 if server i serves at an exponential rate λi,...

In Example $5.3$ if server $i$ serves at an exponential rate $\lambda_{i}, i=1,2$, show that
$$
P\{\text { Smith is not last }\}=\left(\frac{\lambda_{1}}{\lambda_{1}+\lambda_{2}}\right)^{2}+\left(\frac{\lambda_{2}}{\lambda_{1}+\lambda_{2}}\right)^{2}
$$

Key Concepts

Exponential Distribution

The exponential distribution is a continuous probability distribution frequently used to model the time between events in a process where events occur continuously and independently at a constant rate. In service systems, it characterizes the random service times of servers with rates that determine the speed of service completions.

Memoryless Property

A defining feature of the exponential distribution is its memoryless property, which implies that the probability of an event occurring in the future is independent of the time already elapsed. This characteristic greatly simplifies the analysis of sequential events in queuing systems and competing service processes.

Competing Exponentials

When multiple independent exponential processes are in operation, such as servers racing to complete service, the event that occurs first is determined by a competition among these processes. The probability that a particular process triggers the event first is proportional to its rate parameter, forming the basis for calculating associated probabilities in the system.

Order Statistics

Order statistics deal with the arrangement of a sample in increasing order, and in service systems, they are used to analyse the sequence in which events occur. In the context of competing exponential service times, order statistics help in determining the probability that a particular customer (such as Smith) is not the last to be served, by considering the relative order of service completions.

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