The number of arrivals per period is distributed according to the poisson with the 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?
Expected number of units per period(λ)=0.75unit per period
Cost of waiting per period(C_W)=$3.20
Facility cost per unit per period(C_f)=$5.15
The formula for calculating expected service rate is as follows
= λ+√((λC_W)/C_f )
= 0.75 + √([(0.75)($3.20)]/$5.15) = 1.4326 units per period
TC_m = [($3.20 × 0.75)/(1.455 – 0.75)] + ($5.15 × 1.455) = $10.8975
Why did the statistician bring a ladder to the bar?
Because the exponential distribution can be a little tricky to understand, so he needed some help getting to the bottom of it!
To determine the expected service rate, we need to calculate the utilization factor (ρ) using the formula:
ρ = λ / μ
Where:
λ = Arrival rate (expected value of arrivals per period)
μ = Service rate (expected value of service per period)
Given that the expected value of arrivals per period is 0.75 units, we have λ = 0.75.
We also know that the service duration is distributed exponentially, which implies that the expected value of service time (μ) is equal to the reciprocal of the exponential service rate.
Let's calculate the utilization factor (ρ) and the expected service rate:
1. Utilization factor (ρ):
ρ = λ / μ
2. Expected value of service rate (μ):
Since the service time is exponentially distributed, we can use the reciprocal of the exponential service rate.
Note that the exponential service rate is equal to the reciprocal of the expected service time (μ).
Now, let's calculate μ:
μ = 1 / E[T], where E[T] represents the expected value of the service time.
Given that we know the cost of waiting per unit per period ($3.20) and the facility cost for serving one unit per period ($5.15), we can calculate the expected total system cost using the following formula:
Expected Total System Cost = (Cost of Waiting per Unit per Period × Expected Wait Time per Unit) + Facility Cost per Unit
To calculate the expected wait time per unit, we'll use Little's Law:
Expected Wait Time per Unit = Expected Number of Units in the System / Arrival Rate
Now, let's calculate the expected service rate and the expected total system cost step by step:
Step 1: Calculate Utilization Factor (ρ):
Utilization Factor (ρ) = λ / μ
Step 2: Calculate the Expected Service Rate (μ):
Expected Service Rate (μ) = 1 / E[T]
Step 3: Calculate the Expected Total System Cost:
Expected Number of Units in the System (L) = ρ / (1 - ρ)
Expected Wait Time per Unit = L / λ
Expected Total System Cost = (Cost of Waiting per Unit per Period × Expected Wait Time per Unit) + Facility Cost per Unit
Now, let's substitute the given values and calculate the expected service rate and the expected total system cost.
To determine the expected service rate and the expected total system cost, we need to calculate a few key values. Let's break down the steps:
Step 1: Calculate the expected arrival rate (λ) using the given expected value.
λ = expected value = 0.75 units per period
Step 2: Calculate the expected service rate (μ) based on the exponential distribution.
The exponential distribution has a rate parameter (λ) that is the inverse of the expected service time, so we can calculate μ using the following formula:
μ = 1 / expected service time
Step 3: Calculate the traffic intensity (ρ).
The traffic intensity is the ratio of the arrival rate to the service rate:
ρ = λ / μ
Step 4: Calculate the expected waiting time in the system (Wq).
The expected waiting time in the system is given by the following formula:
Wq = ρ / (μ - λ)
Step 5: Calculate the expected number of units in the system (L).
The expected number of units in the system is given by the following formula:
L = λ * Wq
Step 6: Calculate the expected total system cost (C).
The expected total system cost includes the cost of waiting and the facility cost. It can be calculated as follows:
C = (λ * cost of waiting) + (λ * L * facility cost)
Now let's proceed with the calculations:
Step 1: λ = 0.75 units per period (given)
Step 2: To calculate μ, we need the expected service time. However, it is not provided in the question. Please provide the expected service time in order for us to continue the calculation.