Question

In: Advanced Math

How do you mathematically prove a line moves faster than the other? The problem says there...

How do you mathematically prove a line moves faster than the other? The problem says there are two fast food restaurants, each containing two employees working the register. Restaurant 1 has two lines to service customers, and restaurant 2 has one line for customers to wait in. How can you mathematically prove that either restaurant 1 or restaurant 2 has a better line control system?

Side note: I was thinking that this is like which function has the better growth such as an exponential line vs. a power function line, but maybe I'm wrong. Please show me the easiest way.

Expert Solution

This is can shown by using Concepts in Queuing Theory. We can assume poisson arrivals with rate and exponential service rate . Response time of a queue is average time a customer spends in the system (time spent in queue + service time). Call the ratio . This ratio needs to be lesser than one for a stable system. A queue moves faster than other if the Response time is lesser than the other.

We analyze queue A. Restaurant A has two servers and two queues. Restaurant A is essentially two M/M/1 queues. Hence arrival rate for each of the queue is . Response time for a M/M/1 queue is . Hence in this case it is .

Restaurant B has two servers and one queue. Restaurant B is M/M/2 queue. Arrival rate of the queue is . Response time in this case is:

where can be found out to be . Hence,

Consider

We see that since , this ratio is greater than 1, i.e., Response time of restaurant B is lesser than that of restaurant A, i.e., your line moves faster in restaurant B than restaurant A.

Colby Messinger answered 1 week ago

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