Objective of Today’s Lecture
▪ By the end of this lecture you should:
– Understand and identify:
➢ MMS models with finite queue length
➢ MMS models with finite population
– be able to
➢ Identify the computation requirements for realistic cases
➢ Collect the relevant calculations from a Queueing Template spreadsheet
➢ Perform cost analyses.
■The EFIM Manufacturing Company has a group of six identical machines; each machine operates an average of hours between brea kdowns.
■One maintenance operative looks after the machines and takes an average of 2 hours to repair a broken machine
W- What is the average amount of time a machine is out of action once it brea ks down?
– On average, how many broken machines are waiting for the attention of the maintenance operative?
(It can be shown that P。= 0.4845 – can you show this?)
- Queueing Model: M/M/1/w1 6/ EIFO
- λ=1/20 per hour. = 0.05 / hour / machine
– (So Total Breakdown rate 6λ= 0.3 / hour )
- μ= 1/2=0.5 / hour
- λ/ μ=0.05 / 0.5 = 0.1 (frequent term in formulae)
- Need W and Lq
Registration day at a certain university involves two parts: all new students have to go to reception and after this they are sent to their respective course officers depending on which degree they are enrolling for: Management, Business Studies or Economics. 50% of students are BS, 25% are Management and 25% are Economics. The service rates of the three course officers are, on average, 10 students per hour and for the receptionist it is 25 students per hour. If the arrival rate of students at reception is 20 per hour,calculate the average waiting time and number waiting for service at each service point.
The customer support hotline for Bitway Computers is currently staffed by a single technician. Calls arrive randomly at a rate of five per hour. The technician can service calls at a rate of seven per hour on average. The president of Bitway has received numerous complaints from customers about the length of time they must wait “on hold” for service when calling the hotline. She wants to determine the average length of time customers currently wait before the technician answers their calls. If the average waiting time is more than five minutes, she wants to determine how many technicians would be required to reduce the average waiting time to two minutes or less.
Comment on Economic Analysis
Bitway will incur some additional costs in going from one to two customer support technicians.
- Question: What would these be?
- However the improved service level provided by the multiserver system should reduce the number of customer complaints and perhaps lead to favourable word-of-mouth advertising and increased business.
- We could quantify these benefits and compare them to the cost of adding a customer support technician.
- Alternatively the President may simply view the addition of the customer support technician as a competitive necessity.
- Suppose Bitway’s telephone system can keep a maximum of five calls on hold at any point in time. If a new call is made to the hotline when five calls are already in the queue, the new call receives a busy signal.
- One way to reduce the number of calls encountering the busy signal is to increase the number of calls that can be put on hold. However, if a call is answered only to be put on hold for a long time, the caller might find this more annoying than receiving the busy signal.
- The President is interested in investigating what effect adding a second technician would have on the number of calls receiving busy signals and on the average time callers must wait before receiving service.
Comment on Solution Template
- Probability that a customer waits refers to the chance that a customer will have to be put on hold rather than being put through to the technician immediately.
- Probability that a customer balks is the chance that a customer gets the busy signal.
- Question: If Pn is the probability of there being n customers in the system at any point in time, what are the two above probabilities in terms of Pn?
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