Students arrive at the Administrative Services Office at an average of one every
ID: 423085 • Letter: S
Question
Students arrive at the Administrative Services Office at an average of one every 24 minutes, and their requests take on average 16 minutes to be processed. The service counter is staffed by only one clerk, Judy Gumshoes, who works eight hours per day. Assume Poisson arrivals and exponential service times.
a. What percentage of time is Judy idle? (Round your answer to 1 decimal place.)
Percentage of time %
b. How much time, on average, does a student spend waiting in line? (Do not round intermediate calculations. Round your answer to 1 decimal place.)
Average time minutes
c. How long is the (waiting) line on average? (Round your answer to 2 decimal places.)
Average length of the waiting line customers
d. What is the probability that an arriving student (just before entering the Administrative Services Office) will find at least one other student waiting in line? (Do not round intermediate calculations. Round your answer to 2 decimal places.)
Explanation / Answer
Following details are provided :
Arrival rate of students ( @ every 24 minutes ) = a = 60/24 = 2.5 students per hour
Service rate of students ( i.e. processing time of requests @ 16 minutes per student) = S = 3.75 students per hour
= ( 1 – a/s) x 100
= ( 1 – 2.5/3.75) x 100
= 0.3333 x 100
= 33.33%
= a/ s x ( s – a) hour
= 2.5/3.75 x ( 3.75 – 2.5 ) hour
= 2.5 / 3.75 x 1.25 hour
= 0.5333 hour
= 31.998 minute ( 32 rounded of)
AVERAGE TIME STUDENT SPENDS WAITING IN LINE = 32 MINUTES
= a^2/ s x ( s – a )
= ( 2.5 x 2.5)/ 3.75 x ( 3.75 – 2.5 )
= ( 2.5 x 2.5) / ( 3.75 x 1.25 )
= 1.33
AVERAGE LENGTH OF THE WAITING LINE = 1.33 CUSTOMERS
= 1 - probability that there are no students waiting in the line
Probability that there are no students waiting in the line = Po = 1 – a/s = 1 – 2.5/3.75 = 0.333
Therefore ,
Probability that at least one student waiting in the line
= 1 – 0.333
= 0.667( 0.67 rounded to 2 decimal places )
PROBABILITY THAT AN ARRIVING STUDENT WILL FIND AT LEAST ONE OTHER STUDENT WAITING IN THE LINE = 0.667