Question $1$: During lunchtime, customers arrive at a postal office at a rate of $=36$ per hour. The interarrival time of the arrival process can be approximated with an exponential distribution. Customers can be served by the postal office at a rate of $=45$ per hour. The system has a single server. The service time for the customers can also be approximated with an exponential distribution.

Determine the utilization factor.

Group of answer choices

$=45$

$=54$

$=23$

$=45-36=9$

None of the above

Question: $2$ What is the probability that at most $4$ customers arrive within a $5-$minute period? You can use Excel to calculate P$($X$<=$x$).($Please round your answer to four decimals$)$

Question $3$: What is the probability that the service time will be less than or equal to $30$ seconds? You can use Excel to calculate P$($T$<=$t$).($Round your answer to four decimals$)$

Question $4$: Determine the probability that the system is idle, i$.$e$.,$ no customer is waiting or being served.

Group of answer choices

$0=45$

$0=54$

$0=15$

$0=19$

None of the above

Question $5$: Determine the probability that exactly one customer is in the system, i$.$e$.,$ no customer is waiting but one is served.

Group of answer choices

$1=19$

$1=25$

$1=15$

$1=425$

None of the above

Question $6$: Determine the probability that exactly one customer is waiting.

Group of answer choices

$2=281$

$2=16125$

$2=4125$

$2=425$

None of the above

Question $7$: Determine the expected length of the queue, i$.$e$.,$ the number of customer waiting in the queue.

Group of answer choices

$=4$

$=165$

$=125$

$=185$

None of the above

Question $8$: Determine the expected number of customers in the system including the one being served and the ones waiting in the queue.

Group of answer choices

$=4$

$=165$

$=125$

$=185$

None of the above

Question $9$: Determine the expected waiting time, i$.$e$.,$ time spent in the queue, for each customer.

Group of answer choices

$=15$

$=14$

$=445$

$=19$

None of the above

Question $10$: Determine the expected time spent in the system including the time waiting and being served for each customer.

Group of answer choices

$=15$

$=14$

$=145$

$=19$

None of the above