Business continues to grow at Clean Machine. Melanie has
decided to use a second car washing bay, staff ed with another
identical two-person team. Clean Machine will now use a singleline, multiserver, single-phase waiting line system. Th e arrival
rate is estimated to average 24 customers per hour according to
a Poisson distribution. Each of the car wash teams can service an
average of 16 customers per hour according to an exponential
distribution. Calculate the following operational characteristics:
(a) Average system utilization
(b) Average number of customers in the system
(c) Average number of customers waiting in line
(d) Average time a customer spends in the system
(e) Average time a customer spends waiting in line
(f) Probability of having more than four customers in the system
System serving capacity is based on the number of server or systems available in the company. This question is of waiting for model and queuing theory. Here we have arrival rate per hour and service rate per hour. To provide better service it is the aim of every company but customer avoids waiting in line for taking service.
Answer:
In this question, we can calculate the average system utilization rate with help of formula. Average system utilization means busyness of server i.e. how much service is provided by each server.
A.AVERAGE SYSTEM UTILIZATION:
We have arrival rate = 24 customers per hour,$\left(\lambda \right)$
Service rate = 16 customers per hour,$\left(\mu \right)$
$\text{Averagesystemutilization}=\frac{\lambda}{\mu}\phantom{\rule{0ex}{0ex}}=\frac{24}{16}\phantom{\rule{0ex}{0ex}}=1.5$
B.AVERAGE NUMBER OF CUSTOMER IN THE SYSTEM:
$\text{Averageno.ofcustomersinsystem}=\frac{\lambda}{\mu -\lambda}\phantom{\rule{0ex}{0ex}}=\frac{24}{16-24}\phantom{\rule{0ex}{0ex}}=3$
C.AVERAGE NUMBER OF CUSTOMER WAITING IN LINE:
$\text{Averageno.ofcustomerwaitinginline}\left({L}_{Q}\right)=pL\phantom{\rule{0ex}{0ex}}=1.5\times 3\phantom{\rule{0ex}{0ex}}=4.5$
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