Question

Three types of customers arrive at a service station. The times required to service type 1 and type 2 customers are exponential random variables with respective means 1 and 10 seconds. Type 3 customers require a constant service time of 2 seconds. Suppose that the proportion of type 1, 2 and 3 customers is 1/2, 1/8 and 3/8, respectively. Find the probability that an arbitrary customer requires more than 15 seconds of service time.

Answer #1

Customers arrive at bank according to a Poisson process with
rate 20 customers per hour. The bank lobby has enough space for 10
customers. When the lobby is full, an arriving customers goes to
another branch and is lost. The bank manager assigns one teller to
customer service as long as the number of customers in the lobby is
3 or less. She assigns two tellers if the number is more than 3 but
less than 8. Otherwise she assigns...

Customers arrive at
random times, with an exponential distribution for the time between
arrivals. Currently the mean time between customers is 6.34
minutes. a. Since the last customer arrived, 3 minutes have gone
by. Find the mean time until the next customer arrives.
b. Since the last
customer arrived, 10 minutes have gone by. Find the mean time until
the next customer arrives.

6. Consider a queueing system having two servers and no queue.
There are two types of customers. Type 1 customers arrive according
to a Poisson process having rate ??, and will enter the system if
either server is free. The service time of a type 1 customer is
exponential with rate ??. Type 2 customers arrive according to a
Poisson process having rate ??. A type 2 customer requires the
simultaneous use of both servers; hence, a type 2 arrival...

Case Study: Pantry Shop (modified)
Customers arrive at the Pantry Shop store at a rate of 3 per
minute and the Poisson distribution accurately defines this rate. A
single cashier works at the store, and the average time to serve a
customer is 15 seconds, and the exponential distribution may be
used to describe the distribution of service times.
What are λ and μ in this situation?
Using Kendall notation, what type of queuing system is
this?
How many minutes...

Exercise 11.2.5 Customers arrive at Bunkey’s car wash service at
a rate of one every 20 minutes and the average time it takes for a
car to proceed through their single wash station is 8 minutes.
Answer the following questions under the assumption of Poisson
arrivals and exponential service.
(a) What is the probability that an arriving customer will have
to wait?
(b) What is the average number of cars waiting to begin their
wash?
(c) What is the probability...

This is the random process problem.
Vehicles of two different types, cars and trucks, arrive to a
gas station, so that gaps between their arrivals are independent
exponential random variables with parameter 1 (vehicle per hour).
Each vehicle, independently of others, is a car with probability p
and is a truck with probability 1−p. Independently of other
vehicles, each car and truck fills up by the number of gallons that
is uniformly distributed between [8, 12] and [14, 16],
respectively....

Customers arrive at a two-server system according to a Poisson
process having rate λ = 5. An arrival finding server 1 free will
begin service with that server. An arrival finding server 1 busy
and server 2 free will enter service with server 2. An arrival
finding both servers busy goes away. Once a customer is served by
either server, he departs the system. The service times at server i
are exponential with rates µi, where µ1 = 4,
µ2...

A new full-service, attendant-staffed fuel and car service
station on Interstate 95 will service only northbound cars. The
manager estimates that customers will arrive every 4 minutes and
will require 6 minutes to be served at the pump. How many pumps
should be installed if the manager desires a utilization factor of
0.75? What percent of the time will the pumps be idle? Assume that
the coefficients of variation of both the inter-arrival times and
the service times are equal...

A new full-service, attendant-staffed fuel and car service
station on Interstate 95 will service only northbound cars. The
manager estimates that customers will arrive every 4 minutes and
will require 6 minutes to be served at the pump. How many pumps
should be installed if the manager desires a utilization factor of
0.75? What percent of the time will the pumps be idle? Assume that
the coefficients of variation of both the inter-arrival times and
the service times are equal...

Customers arrive to a single server system in accordance with a
Poisson pro- cess with rate λ. Arrivals only enter if the server is
free. Each customer is either a type 1 customer with probability p
or a type 2 customer with probabil- ity 1 − p. The time it takes to
serve a type i customer is exponential with rate μi , i = 1, 2.
Find the average amount of time an entering customer spends in the
system.

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