Customers arrive at a suburban ticket outlet at the rate of 14 per hour on Monday mornings. This can be described by a Poisson distribution. Selling the tickets and providing general information takes an average of 3 minutes per customer, and varies exponentially. There is 1 ticket agent on duty on Mondays. Determine each of the following:______________(A) system utilization.(B) average number in line.(C) average time in line.(D) average time in the system.

When the arrival rate matches the distribution of Poisson, the rate of delivery varies exponentially as well as the server exists. The paradigm, therefore, is W/M/I, an exponential time interval concept, and one server.

Arrival rate [tex](\lambda) = 14\ per\ hour[/tex]

Service rate[tex]( \mu) =\frac{3\ minutes}{customer} = \frac{60 minutes (1 hour)}{3 minutes}= 20 \ \frac{customers}{hour}[/tex]

For point A:

System utilization[tex]=\frac{\lambda}{\mu}=\frac{14}{20}=0.7[/tex]

For point B:

Average number in line:

[tex](L_q )=\frac{\lambda^2}{\mu \times (\mu-\lambda)}\\\\=\frac{14^2}{20\times (20-14)}\\\\= 1.633[/tex]

For point C:

Average time in line:

[tex](W_q)=\frac{L_q}{\lambda}\\\\=\frac{1.633}{14} \ hours\\\\=\frac{1.633}{14} \times 60 \ minutes\\\\= 7\ minutes[/tex]

For point D:

Average time in the system:

[tex](W_s)=w_q+\frac{1}{\mu}\\\\=7\ minutes + \frac{1}{20} \ hours\\\\=7 \ minutes + \frac{60}{20} \ hours\\\\= 7 \ minutes + 3\ minutes\\\\= 10 \ minutes\\[/tex]