Question

The city council of a small town has decided to build a tennis court in the central park. Players are expected to arrive on the average of 10 sets of players per 12-hour day. Playing time is exponentially distributed with a mean of one hour. Arrivals are Poisson. However, a member of the city council contends that people will not wait if two groups are already waiting. Determine the necessary statistical data and make a recommendation based on these.

Answer 1

Given are following data :

Arrival rate of players = a = 10 / day

Service rate ( i.e. playing time of 1 hours) = S = 12/1 = 12

Probability that ZERO people will be waiting =Po = ( 1 – a/s ) = 1 – 10/12 = 1 – 0.833 = 0.167

Probability that 1 person will be waiting = P1 = ( a/s ) x PO = ( 10/12) . Po = 0.833 x 0.167 = 0.139

Hence, probability that less than 2 people will be waiting

= Probability that ZERO people are waiting + Probability that 1 person will be waiting

= 0.167 + 0.139

= 0.306

Therefore, probability that two or more people will be waiting

= 1 – Probability that less than 2 people will be waiting

= 1 – 0.306

= 0.694

Chances ( probability ) that people will not wait therefore is 0.694. Since this is a reasonable high probability, effort should be made to reduce value of it.