The number of arrivals per period is distributed according to the Poisson with an expected value of 0.75 unit per period. The cost of waiting per unit per period is $3.20. The facility cost for serving one unit per period is $5.15. What expected service rate should be established if the service duration is distributed exponentially? What is the expected total system cost?
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To find the expected service rate, we need to calculate the average service time (also known as the service rate) using the expected value of the Poisson distribution.
Step 1: Calculate the average service time.
Since the service time follows an exponential distribution, the average service time (µ) is the inverse of the expected value of the Poisson distribution.
Given that the expected value of the Poisson distribution is 0.75 units per period, we can calculate the average service time as follows:
µ = 1 / (expected value of Poisson distribution) = 1 / 0.75 = 1.33 units per period.
So, the average service time is 1.33 units per period, which can also be interpreted as the service rate.
Step 2: Calculate the expected total system cost.
The total system cost can be calculated as the sum of the waiting cost and the facility cost per unit.
Given:
- The cost of waiting per unit per period = $3.20
- The facility cost for serving one unit per period = $5.15
Since the number of arrivals is Poisson distributed with an expected value of 0.75 units per period, the expected waiting cost can be calculated as follows:
Expected waiting cost = (average waiting time) x (cost of waiting per unit per period)
To find the average waiting time, we need to calculate the traffic intensity (ρ) using the formula:
ρ = (arrival rate) x (average service time)
Arrival rate = expected value of the Poisson distribution = 0.75 units per period.
Average service time = 1.33 units per period.
ρ = 0.75 x 1.33 = 0.9975.
Next, we calculate the average number of customers waiting in the system using the formula:
Average number of customers waiting in the system = ρ^2 / (1 - ρ).
Average number of customers waiting in the system = (0.9975)^2 / (1 - 0.9975) = 0.997246.
Finally, the average waiting time can be calculated using Little's Law:
Average waiting time = (average number of customers waiting in the system) / (arrival rate).
Average waiting time = 0.997246 / 0.75 = 1.3297 period.
Now, we can calculate the expected waiting cost:
Expected waiting cost = (average waiting time) x (cost of waiting per unit per period) = 1.3297 x $3.20 = $4.255.
The expected total system cost is the sum of the waiting cost and the facility cost per unit:
Expected total system cost = Expected waiting cost + (facility cost for serving one unit per period) = $4.255 + $5.15 = $9.405.
Therefore, the expected service rate should be established as 1.33 units per period, and the expected total system cost is $9.405.