Process Management Course Work: Arrival Rate and Service Rate
QUESTION
The university library has to decide on the most optimal configuration for the student copy machines. Student demand for making copies peak during the evening 6pm-12 midnight hours and the library wants to minimize the wait times for the students while not investing more than necessary in the copy machines. The options for the service configurations are as follows:
1) Use a single copy machine that can make a maximum of 300 copies per hour. Due to the variability in the size and nature of the jobs, an average job is completed anywhere between 1 minute to 10 minutes (assume a uniform distribution).
2) Use two copy machines each of which can make a maximum of 100 copies per hour. Since these are slower machines compared to that in #1, the service times follow a uniform distribution between 3 minutes and 20 minutes. The options for the queue configurations are either to have a single queue (for either of the above two options) or have two dedicated queues (for option #2). The time between arrivals of students to the copy machine system is approximately one student every 4 minutes. Assume this inter-arrival time follows an exponential distribution.
Looking at the options, make the following decisions:
1) Can either configuration sustain the arrival rates of the students? Give reasons.
2) Which configuration seems to be the better option to adopt? Why?
3) Would your decision on the above change and how if the following happens:
- The inter-arrival time changes to a triangular distribution between 1 minute and 6 minutes with the most likely time being 2 minutes?
- The service times change to exponential distributions (mean time of 5 minutes with the single machine option #1 and 10 minutes with the two-machine option #2)
4) For all analysis, use appropriate models and solution approached to determine your answers.
Write a short executive report on the above decisions and findings, and provide all the calculations, models and analysis in the appendices. Submit any supplemental files that you may have used for answering the question.
Process management coursework
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1] Can either configuration sustain the arrival rates of the students? Give reasons.
λ mean arrival rate
µ- mean service rate
The queues tend to infinity when the arrival rate (lambda) is greater or equal than 1 times service rate per server (mu). Under option 1 the service rate of the photocopying machine is 300 copies per hour. This implies that that the students should ideally have less than 300 copies to be made since this is what can be handled by the single photocopier. The absolute minimum number of server increases when the arrival rate of the students is more than the single photocopy machine can handle. The worst level of service requires 2 servers under Queuing rule of thumb.
Under option 2 the copy machines can handle 100 copies each at 200 copies per hour for the two photocopiers. However, the inter arrival time is 4 minutes, and for the queuing systems to work properly then the service rate is between 3 minutes to 20 minutes. As such, it is difficult to sustain the arrival rates in this option where the machines have slow efficiency rates with service between 3 and 20 minutes, and the average service time is 12 minutes. For the configuration to meet the arrival rates then they should arrive be 12 minutes or lower.
2] Which configuration seems to be the better option to adopt? Why?
Option 1 is the better option between the two assuming that the average service rate is 5 minutes given that it requires between 1 to 10 minutes completing the job. On the other hand, the inter arrival time is 4 minutes. The average time waiting in line λ/ µ is 4/5= 0.8 and this also represents the queuing utilization is 80%. On the other hand, the queue length in the system/ average number of students in the system is 4 being λ/ µ- λ= 4(5-4). The first option is also more efficient highlighting it should be chosen, but a breakdown of the machine would halt the photocopying operations.
3] Would your decision on the above change and how if the following happens:
The inter-arrival time changes to a triangular distribution between 1 minute and 6 minutes with the most likely time being 2 minutes.
A change in the inter arrival indicates that the utilization rate of the service is lower since the service rate still remains the same. The probability of the idle server also increases, while the average time spent in the system reduces (delay in t...
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