Questions: Cuntomers artive at a Tim Hortons Drive-through at a rate of 10 every hour on average. The senvice time in 4 min on average. Analyse the performance of this drive through.

Cuntomers artive at a Tim Hortons Drive-through at a rate of 10 every hour on average. The senvice time in 4 min on average. Analyse the performance of this drive through.
Transcript text: Cuntomers artive at a Tim Hortons Drive-through at a rate of 10 every hour on average. The senvice time in 4 min on average. Analyse the performance of this drive through.
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Solution

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Solution Steps

Step 1: System Utilization

The system utilization (\( \rho \)) is calculated as the ratio of the arrival rate (\( \lambda \)) to the service rate (\( \mu \)): \[ \rho = \frac{\lambda}{\mu} = \frac{10}{15} = 0.6667 \]

Step 2: Probability the System is Empty

The probability that the system is empty (\( P_0 \)) is given by: \[ P_0 = 1 - \rho = 1 - 0.6667 = 0.3333 \]

Step 3: Average Number in Line

The average number of customers in line (\( L_q \)) is calculated using the formula: \[ L_q = \frac{\lambda^2}{\mu(\mu - \lambda)} = \frac{10^2}{15(15 - 10)} = 1.3333 \]

Step 4: Average Number in System

The average number of customers in the system (\( L \)) is: \[ L = L_q + \frac{\lambda}{\mu} = 1.3333 + \frac{10}{15} = 2.0 \]

Step 5: Average Time in Line

The average time a customer spends in line (\( W_q \)) is: \[ W_q = \frac{L_q}{\lambda} = \frac{1.3333}{10} = 0.1333 \text{ hours} \]

Step 6: Average Time in System

The average time a customer spends in the system (\( W \)) is: \[ W = W_q + \frac{1}{\mu} = 0.1333 + \frac{1}{15} = 0.2 \text{ hours} \]

Final Answer

The performance analysis of the Tim Hortons Drive-through yields the following results:

  • System Utilization: \( \rho = 0.6667 \)
  • Probability the System is Empty: \( P_0 = 0.3333 \)
  • Average Number in Line: \( L_q = 1.3333 \)
  • Average Number in System: \( L = 2.0 \)
  • Average Time in Line: \( W_q = 0.1333 \) hours
  • Average Time in System: \( W = 0.2 \) hours

Thus, the final boxed answers are: \[ \boxed{\rho = 0.6667, \quad P_0 = 0.3333, \quad L_q = 1.3333, \quad L = 2.0, \quad W_q = 0.1333, \quad W = 0.2} \]

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