Suppose vehicles arrive at a single toll booth according to a Poisson process with a mean...
Suppose vehicles arrive at a single toll booth according to a Poisson process with a mean arrival rate of 8.4 veh/min. Their service times are exponentially distributed. The mean processing rate is 10 veh/min.
a. What is the value of the utilization ratio?
b. What is the average length of the queue?
c. What is the average waiting time in the queue?
d. What is the average time spent in the system?
Homework Answers
The solution is based on the queuing theoy.
Given data:-
mean arrival rate (
)=
8.4veh./min.
mean service rate (
)=10veh./min.
Solutions:-
(a)utilization rate :-
formula for utili0ation rate =
i.e. u.r.=8.4/10 =.0.84
(b)avg. length of queue(Lq):-
formula for length of queue in terms of vehicles=
i.e. Lq=8.42/10(10-8.4)=4.41veh.
(c)avg. waiting time in the queue(Wq) :-
Formula =
Lq=8.4/10(10-8.4)=0.52 min.=31.5 seconds
(d) avg. time spent in the system(Ws) :-
Formula=
Ws=1/1.6=0.625 minutes=37.5 seconds
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