04. More Complex Markov Processes

# 04. More Complex Markov Processes - CS4 Modelling and...

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CS4 Modelling and Simulation LN-4 4 More Complex Markov Processes 4.1 Introduction In this note we consider another example Markov process, modelling a more realistic system than those considered in the previous lecture notes. However we will quickly see that even for such moderately sized examples diﬃculties soon arise when we work directly at the level of the Markov process. Several attempts may be necessary before a correct characterisation of the state space is achieved; generating the state space and calculating the transition rates is time consuming and error prone; interpreting the results and deriving the performance measures is also a complex task. The last two problems are helped by using a high-level modelling paradigm, as we will discuss in the last section of the note. 4.2 Upgrading a PC LAN 4.2.1 Description of the system The problem we consider is to determine the mean waiting time for data packets at a PC connected to a local area network, operating as a token ring. Such a network uses a transmission medium that supports no more than one transmission at any given time. To resolve conﬂicts, a token is passed round the network from one node to another in round robin order. A node has control of the medium, i.e. it can transmit, only whilst it holds the token. In a PC LAN every PC corresponds to a node on the network. Other nodes on the network might be peripheral devices such as printers or faxes but for the purposes of this study we make no distinction and assume that all nodes are PCs. There are currently four PCs (or similar devices) connected to the LAN in a small oﬃce, but the company has recently recruited two new employees, each of whom will have a PC. Our task is to find out how the delay experienced by data packets at each PC will be affected if another two PCs are added. Each PC can only store one data packet waiting for transmission at a time, so at each visit of the token there is either one packet waiting or no packet waiting. The average rate at which each PC generates data packets for transmission is known to be λ (see Figure 2). We also know the mean duration, d , of a data packet transmission, and the mean time, m , taken for the token to pass from one PC to the next. It is assumed that if another data packet is generated, whilst the PC is transmitting, this second data packet must wait for the next visit of the token before it can be transmitted. In other words, each PC can transmit at most one data packet per visit of the token. 4.2.2 Modelling the system The first stage of developing a Markov process to represent the system is to choose the random variables which will be used to characterise the states of the system. Firstly, considering the nodes, we can think of each PC as being in one of two states: the PC does not currently have a data packet to transmit; or 24

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CS4 Modelling and Simulation LN-4 the PC has a data packet waiting at the interface for transmission.
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