CONTINUOUS-TIME MARKOV CHAINS in .NET framework

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(c) What is the long-run average number of cars in the station (d) What is the long-run fraction of potential customers that are lost 45 A production hall contains a fast machine and a slow machine to process incoming orders Orders arrive according to a Poisson process with rate An arriving order that nds both machines occupied is rejected Unless both machines are occupied, an arriving order is assigned to the fast machine if available; otherwise, the order is assigned to the slow machine The processing time of an order is exponentially distributed with mean 1/ 1 at the fast machine and mean 1/ 2 at the slow machine It is not possible to transfer an order from the slow machine to the fast machine (a) Formulate a continuous-time Markov chain to analyse the situation in the production hall Specify the state variable(s) and the transition rate diagram (b) Specify the equilibrium equations for the state probabilities What is the long-run fraction of time that the fast (slow) machine is used What is the long-run fraction of incoming orders that are lost 46 In Gotham City there is a one-man taxi company The taxi company has a stand at the railway station Potential customers arrive according to a Poisson process with an average of four customers per hour The taxi leaves the station immediately a customer arrives A potential customer nding no taxi present waits until the taxi arrives only if there are less than three other customers waiting; otherwise, the customer goes elsewhere for alternative transport If the taxi returns to the stand and nds waiting customers, it picks up all waiting customers and leaves The amount of time needed to return to the stand has an exponential distribution with mean 1/ i when the taxi leaves the stand with i customers, i = 1, 2, 3 (a) Formulate a continuous-time Markov chain to analyse the situation at the taxi stand Specify the state variable(s) and the transition rate diagram (b) What is the long-run fraction of time the taxi waits idle at the taxi stand What is the long-run fraction of potential customers who go elsewhere for transport 47 A container terminal has a single unloader to unload trailers which bring loads of containers The unloader can serve only one trailer at a time and the unloading time has an exponential distribution with mean 1/ 1 After a trailer has been unloaded, the trailer leaves but the unloader needs an extra nishing time for the unloaded containers before the unloader is available to unload another trailer The nishing time has an exponential distribution with mean 1/ 2 A leaving trailer returns with the next load of containers after an exponentially distributed time with mean 1/ There are a nite number of N unloaders active at the terminal (a) Formulate a continuous-time Markov chain to analyse the situation at the container terminal Specify the state variable(s) and the transition rate diagram (b) What is the long-run fraction of time the unloader is idle What is the long-run average number of trailers unloaded per time unit (c) What is the long-run average number of trailers waiting to be unloaded What is the long-run average waiting time per trailer (d) Write a computer program to compute the performance measures in (b) and (c) for the numerical data N = 10, 1 = 1/3, 2 = 2 and = 1/50 48 Messages for transmission arrive at a communication channel according to a Poisson process with rate The channel can transmit only one message at a time The transmission time is exponentially distributed with mean 1/ The following access control rule is used A newly arriving message is accepted as long as less than R other messages are present at the communication channel (including any message in transmission) As soon as the number of messages in the system has dropped to r, newly arriving messages are again admitted to the transmission channel The control parameters r and R are given integers with 0 r < R (a) Formulate a continuous-time Markov chain to analyse the situation at the communication channel Specify the state variable(s) and the transition rate diagram
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