Question

At the order fulfillment center of a major mail-order firm, customer orders, already packaged for shipment, arrive at the sorting machine to be sorted for loading onto the appropriate truck for the parcel's address. The arrival rate at the sorting machine is at the rate of 100 per hour following a Poisson distribution. The machine sorts on average at the rate of 125 per hour and exponentially distributed. (a) What is the average number of packages waiting to be sorted? (b) What is the average number of packages in the sorting system? (c) How long must the average package wait until it gets sorted?

Answer 1

This is a single queue with Poisson arrival and exponential service rate question. Hence, above formulas will be used.

1. Using the Formula 1 in the image,

Average number of packages waiting to be sorted = 100² / 125(125-100) = 3.2 packages

2. Using the formula 2 in the image,

Average number of packages in the system = 100 / 125-100 = 4 packages

3. Using the formula 3 in the image,

Average time a package spends in the queue = 100 / 125 x ( 125 - 100) = 0.032 hours = 115.2 seconds