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Homework Statement
There are two queues at a checkout in a grocery store. You are in Q1, behind 2 people (one of them is just starting to be served). Your friend is in Q2, behind 1 person (who just started to be served). The expected service time is \lambda minutes. The service at each checkout is independent, identical, and exponentially distributed.
(a) What is the expected time until you check out?
(b) What is the probability that your friend is still waiting when you start to be serviced at the checkout?
Homework Equations
Mean of exponential distribution is [itex] \frac{1} { \lambda } [\itex], whenever the parameter is \lambda.
The PDF for an exponential distribution is: p(t) = \lambda e^{-\lambda t}
The CDF for an exponential distribution is: P(t) = 1 - e^{- \lambda t}
The Attempt at a Solution
(a) I think this is [itex]3 \lambda[\itex], because of the memoryless property. Not sure how to formalize this, though.
(b) I was thinking that I should use the CDF here. Suppose that Q1 is composed of person A and person B. Person A finishes at time s. Person B finishes at time t, t > s. Q2 is composed of person C, who does not finish by time t.
[tex] Pr ($Q1 done by time $t) = Pr($B done by time $t - s) \cdot \int_0^t {Pr(A done by time s)ds}[\tex]
And then Pr(our event) = integral over t from 0 to infinity {Pr(Q1 done by time t) Pr(C not done by time t)}
Using the CDF in all cases. Is this the correct approach?
P.S. How do I write in Latex on these forums?
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