1)A club has two services desks; one
at each entrance of the store. Customer arrives at each desk at the average of
one every 6 minutes.
The service rate is 4 minutes per
customer;
The club is not considering consolidating the
two service desks into one location staffed by two clerks who will be working
at the same rate.
a) What is the probability of waiting in line?
b) How many customers are waiting in line?
c) how much time a customer spend at a service desk?(waiting +service time)
d) Do you think the club should consolidate the service desks?
Sol:
λ = Arrival
rate=6mins, μ = Service rate=4*2=8mins
a)
The probability of waiting in line is given by,
Where Lq
= Average number waiting in line, S = Number of service channels=2
= 1.125 customers
Therefore,
= 1.125*((2*8/6)-1) =1.185
b)
The number of customers waiting in line, Lq
= 1.125 customers.
c)
The time spent,
Where Ls= Average
number in system (including any being served) given by,
Ls = 1.125+6/8=1.875
Ws = 1.875/6= 0.3125mins
d)
No, the club should not consolidate the service
desks, since there is a very good customer sharing taking place between the two
service desks.
2) A cafeteria has a coffee urn from which customers serve themselves;
arrival at the urn follow a poisson distribution at the rate 3 per minutes.
customer’s takes about 15 second to serve themselves; exponential distribution.
a) How many customers would you expect to see at the on the average at the
coffee urn?
b) How long would you expect it to take to get a cup of coffee?
c) What percentage of time is the urn being used?
d) What is the probability that three or more customers are in the
cafeteria?
e) If the cafeteria installs an automatic vendor that dispenses a cup of
coffee at a constant of 15 seconds, how does this change your answer to a) and
b)?
Sol:
a)
The number of customers will be
Where λ =
Arrival rate=3/min, μ = Service rate=4/min
Ls = 3 customers
b)
Waiting time,
Where
Lq = 2.25
Wq = 0.75mins
c)
Percentage time= Wq/ Ws
where
Ws=1
Percentage= 75%
d)
Where n=3,
Pn= 0.14
e)
Ls= 4 at max
Therefore
from the above formulae, Lq=3.25
Wq= 1.08mins
The
reason why I used all the above formulae is because, these formulae’s are
called wait line formulae’s, which is mainly, used to calculate the wait times.
It has got four models, Model 1-4. Hence based on the requirements the
corresponding models are used. Note that all the formulae’s belonging to same
model has to be used in one problem.
