# Time study

Topics: Arithmetic mean, Road, Transport Pages: 2 (744 words) Published: September 14, 2014
Homework #4
IOE 416 - Queueing Systems
Homework #4 (Two pages)

Answers are due before start of class on Thursday, April 17th, 2014

1. Fill out the IOE 416 teaching evaluation questionnaire on CTools (4 points out of 20). Note: Attach a copy of the e-mail receipt to your homework to be credited with the 4 points. 2. Items in a manufacturing facility form a queue to be processed at a single workstation. The items arrive at the queue in a Poisson process, and processing times at the workstation are independent with an unspecified distribution, so that the queue operates as an M/G/1 system. The arrival rate to the queue is 20 items per hour, and the mean processing time at the workstation is 2.4 minutes.

It is observed that the number of items in the system, N, has a mean of 2.8 items, and a standard deviation of 2.3 items.
(a) What is the standard deviation of the processing times at the workstation (in minutes)? Let T denote time in system for an item. Using Little’s Law and its extension for an M/G/1 queue, calculate

(b) the mean time in system, W = E[T] (in minutes)
(c) the standard deviation of times in system, StdDev[T] (in minutes). 3. Vehicles in a traffic stream flow in a Poisson process along a lane of main highway with rate 1620 vehicles per hour. A vehicle on a minor road wants to merge with the traffic. The driver of the merging vehicle has a critical gap of 5 seconds (i.e., accepts a gap in the traffic to merge if the gap ≥ 5 seconds). The first gap the driver encounters is the time to the next vehicle in the traffic after the driver arrives at the merge point. The lengths of vehicles on the main highway can be neglected.

(a) What is the probability that the driver of the merging vehicle has no delay? (b) What is the average number of gaps that a driver rejects before being able to merge? (c) What is the driver’s expected delay (in seconds)?

4. On the minor road in Question 2 above, vehicles form a queue to merge into the main traffic, and...