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JOHN RUNS A SMALL OIL CHANGE CENTER WITH ONLY A SINGLE BAY

JOHN RUNS A SMALL OIL CHANGE CENTER WITH ONLY A SINGLE BAY

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Author: Christine Farr
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John runs a small oil change center with only a single bay. On average, a customer arrives at John’s shop every 40 minutes for oil change. Each oil change service takes 30 minutes. Both the customers’ arrival process and the oil change service contain some uncertainties (variability).  The arrival process follows a Poisson process and the service time is exponentially distributed (The system forms a M/M/1 Queue). The coefficient of variation for the arrival process is 1. The coefficient of variation for the service (oil change) process is also 1.
 (a)  What are the arrival rate, service rate and the utilization of the John’s shop. 
       Arrival rate λ =                                               . (0.2 points)
       Please show your analysis below if necessary: 
         
      Service rate μ=                                             . (0.2 points)
      Please show your analysis below: 
 
     Utilization ρ=                         . (0.25 points)
     Please show your analysis below: 
                  
 
(b)  With what fraction of time John’s shop is empty? 
 John’s shop is empty                             % of the time. (0.25 points)
        Please show your analysis below is necessary: 
         
 
 
(c)  If you are a customer of John’s shop, what kind of waiting time will you be concerned with?  (circle one)  (0.2 points)
          Waiting time in queue (not including service time)                   Total waiting time    
          
   Please briefly explain your choice:      
 
 
(d) Based on your choice in (c), calculate the corresponding average waiting time.
The average waiting time is                                     . (0.25 points)
       Please show your analysis below:   
   
 
 
Please study the slides of Lecture 8 and answer questions (e) – (g) 
 (e)  John’s shop locates in the middle of nowhere. Given its location, both the customers under service and the ones waiting to be served choose to stay at the shop during their visits. 
How many customers are there in John’s shop on average? 
       On average, there are                                 customers at John’s shop.   (0.25 points)
Please show your analysis below: 
 
 
 
 (f)  There is a small waiting room at John’s shop with 4 seats. With what probability that there are people waiting at John’s shop without a seat, i.e., there are 5 or more people in the John’s shop?  
With probability                         there are people waiting without a seat. (0.2 points)
       Please show your analysis below: 
 


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