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Chapter 8: Problem 2

Machines in a factory break down at an exponential rate of six per hour. There is a single repairman who fixes machines at an exponential rate of eight per hour. The cost incurred in lost production when machines are out of service is \(\$ 10\) per hour per machine. What is the average cost rate incurred due to failed machines?

Short Answer

Expert verified
The average cost rate incurred due to failed machines is \(\$30\) per hour.

Step by step solution

01

Identify the Failure Rate and Repair Rate

In the problem, we are given the failure rate and repair rate for machines. The failure rate \(\lambda\) is six per hour, and the repair rate \(\mu\) is eight per hour.
02

Calculate Average Number of Failed Machines

Using the fact that the average number of failed machines can be found using the formula \(\frac{\lambda}{\mu-\lambda}\), we can plug in our failure and repair rates: \[\text{Average Number of Failed Machines} = \frac{6}{8-6} = \frac{6}{2} = 3\]
03

Calculate the Average Cost Rate

Now that we have the average number of failed machines, we can calculate the average cost rate. Given that the cost incurred per hour per failed machine is \(\$10\), the average cost rate is: \[ \text{Average Cost Rate} = 3 \cdot 10 = \$ 30 \text{ per hour} \] Thus, the average cost rate incurred due to failed machines is \(\$30\) per hour.

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Most popular questions from this chapter

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 \(\lambda_{1}\), and will enter the system if either server is free. The service time of a type 1 customer is exponential with rate \(\mu_{1}\). Type 2 customers arrive according to a Poisson process having rate \(\lambda_{2}\). A type 2 customer requires the simultaneous use of both servers; hence, a type 2 arrival will only enter the system if both servers are free. The time that it takes (the two servers) to serve a type 2 customer is exponential with rate \(\mu_{2}\). Once a service is completed on a customer, that customer departs the system. (a) Define states to analyze the preceding model. (b) Give the balance equations. In terms of the solution of the balance equations, find (c) the average amount of time an entering customer spends in the system; (d) the fraction of served customers that are type \(1 .\)

Consider a system where the interarrival times have an arbitrary distribution \(F\), and there is a single server whose service distribution is \(G\). Let \(D_{n}\) denote the amount of time the \(n\) th customer spends waiting in queue. Interpret \(S_{n}, T_{n}\) so that $$ D_{n+1}=\left\\{\begin{array}{ll} D_{n}+S_{n}-T_{n}, & \text { if } D_{n}+S_{n}-T_{n} \geqslant 0 \\ 0, & \text { if } D_{n}+S_{n}-T_{n}<0 \end{array}\right. $$

A supermarket has two exponential checkout counters, each operating at rate \(\mu .\) Arrivals are Poisson at rate \(\lambda\). The counters operate in the following way: (i) One queue feeds both counters. (ii) One counter is operated by a permanent checker and the other by a stock clerk who instantaneously begins checking whenever there are two or more customers in the system. The clerk returns to stocking whenever he completes a service, and there are fewer than two customers in the system. (a) Find \(P_{n}\), proportion of time there are \(n\) in the system. (b) At what rate does the number in the system go from 0 to \(1 ?\) From 2 to \(1 ?\) (c) What proportion of time is the stock clerk checking? Hint: Be a little careful when there is one in the system.

There are two types of customers. Type 1 and 2 customers arrive in accordance with independent Poisson processes with respective rate \(\lambda_{1}\) and \(\lambda_{2}\). There are two servers. A type 1 arrival will enter service with server 1 if that server is free; if server 1 is busy and server 2 is free, then the type 1 arrival will enter service with server 2\. If both servers are busy, then the type 1 arrival will go away. A type 2 customer can only be served by server \(2 ;\) if server 2 is free when a type 2 customer arrives, then the customer enters service with that server. If server 2 is busy when a type 2 arrives, then that customer goes away. Once a customer is served by either server, he departs the system. Service times at server \(i\) are exponential with rate \(\mu_{i}\), \(i=1,2\) Suppose we want to find the average number of customers in the system. (a) Define states. (b) Give the balance equations. Do not attempt to solve them. In terms of the long-run probabilities, what is (c) the average number of customers in the system? (d) the average time a customer spends in the system?

Customers arrive at a two-server system according to a Poisson process having rate \(\lambda=5\). 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 \(\mu_{i}\), where \(\mu_{1}=4, \mu_{2}=2\) (a) What is the average time an entering customer spends in the system? (b) What proportion of time is server 2 busy?
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