Gotham City has 10,000 streetlights. City investigators have determined that at any given time, an average of 1,000 lights are burned out. A street light burns out after an average of 100 days of use. The city has hired Mafia, Inc., to replace burned-out lamps. Mafia, Inc.'s contract states that the company is supposed to replace a burned-out street lamp in an average of 7 days. Do you think that Mafia, Inc. is living up to the contract

11.11

Explanation:

λ=1/100/day

K=10,000

Therefore the expected length of time for replacing a burned-out lamp is equally the expected waiting time in the system which is W.

L = 1,000 (average number of burned-out lamps)

Effective arrival rate:

¯λ=λ (K-L) = 1/100 (10,000-1,000) = 90/day

Average length of time it takes to replace a burned-out lamp is:

W=L/¯λ = 1,000/90 = 11.11

Mafia, Inc. is not living up to the contract since the company is supposed to replace the burned-out street lamp in an average of 7 days.

This means that Mafia, Inc. is NOT living up to the contract (

Explanation:

We can relate this question to machine repair problem M/M/1.

Using the formula λ*=λ (K-L)

Where λ is average days of usage = 1/100 per day

K is Total sample of streetlights = 10000

L is the average number of burned-out lamps = 1000

The expected length of time for replacing a burned-out lamp is the expected waiting time given by W = L/λ*

Therefore the effective arrival rate λ*=λ (K-L) = 1/100 (10000-1000)

= 90/day

Thus, the average length of time it takes to replace a burned-out lamp is

W=L/λ * = 1000/90≈11.11 days

This means that Mafia, Inc. is NOT living up to the contract (contract states that the company is supposed to replace a burned-out street lamp in an average of 7 days).