Question

In a super marker the average arrival rate of customers is 5 in every 30 minutes. The arrival time it takes to list and calculate the customers purchase at the cash desk is 4.5min and this time is exponentially distributed.
(a)How long will the customer expect to wait for service at the cash desk
(b)What is the chance that the queue length will exceed 5
(c)What is the probability that the cashier is working

Answer 1

Shouldn't the section "The arrival time it takes to list....." be written as "The average time it takes to list........" ?

Answer 2

theres a lot going on here... is this a computer programming problem?

im not sure how to mathematically set this up without simulating a sample given the exponential distribution, then calculate answers based on average of many simulations

you have people entering the queue at a constant rate of 1 every 6 minutes
the time at the desk varies with a exponential random variable where lambda = 2/9

you can find probability that someone is at desk for more than 6 minutes, thus starting a line
\[P(x>6) = e^{-6 \lambda} \approx 0.2636\]
but then you have the compound problem of multiple people lasting more than 6 minutes which will effect the length of the queue

so i don't know how to set up an equation for knowing length of queue at any given time or how long they have been in queue

i would solve this by modeling the situation in a computer program, generating the random variable for each customer in queue
\[x = \frac{- \ln(random[0,1])}{\lambda}\]
this will model the exponential distribution for the random variable