Question

(Operations Management) A residential construction company has opened a registration office. Customers arrive at the rate of 200 per hour (Poisson Distribution) and the cost of their waiting in the queue is estimated $100 per person per hour. The local convention bureau provides servers to register customers at the fee of $15 per person per hour. Registration process takes 1 minute (Exponential Distribution) and a single waiting line, with multiple servers is set up.
a) Compute both the minimum and optimal number of servers for this system.
b) What is the cost of system (per hour) at the optimum number of servers?
c) Compute the server utilization rate with the minimum number of servers.
d) A new registration manager is hired who initiates a program to entertain the people in line with juggler whom she pays $15/hour. This reduces the waiting cost to $50/hour. What is the optimal number of servers and the cost of that in this case?

Answer 1

Arrival rate, λ = 200 per hr
Service rate, μ = 1 in 1 minute = 60 per hour

a)

The minimum number of servers required > λ/μ i.e. 200/60 or 3.333

So, the minimum number of servers required is 4

Use the two formulae to compute the idle server probability (P0) and average queue length (Lq) for various server numbers (s)

# servers (s)	P0	Lq	Waiting cost = $100 * Lq	Cost of server = $15 * s	Total cost per hour
4	0.0213	3.289	$328.86	$60.00	$388.86
5	0.0318	0.653	$65.33	$75.00	$140.33
6	0.0346	0.185	$18.53	$90.00	$108.53
7	0.0354	0.056	$5.57	$105.00	$110.57

The total cost per hour is minimized for s=6. So, keeping six servers is the optimal decision.

b)

Cost of system (per hour) at the optimum number of servers is $108.53

c)

Utilization with s=4 will be u = λ/s.μ = 200/(4*60) = 83.33%

d)

The effective reduction is waiting cost = 50 - 15 = $35 per hour

# servers (s)	P0	Lq	Waiting cost = $50 * Lq	Cost of server = $15 * s	Cost of juggler	Total cost per hour
4	0.0213	3.289	$164.43	$60.00	$15.00	$239.43
5	0.0318	0.653	$32.67	$75.00	$15.00	$122.67
6	0.0346	0.185	$9.26	$90.00	$15.00	$114.26
7	0.0354	0.056	$2.78	$105.00	$15.00	$122.78

So,

s = 6 is still optimal.

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