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# State Transition Diagrams

Sometimes, it is important to not only have the number of states of each type but also their order. If it is assumed that the next state depends only on the cirrent system state, then the system follows a Markov model. These models are commonly used in queuing analysis. They can be described by a transition matrix, which gives the probabilities of the next state given the current state.

Performance can be modelled and solved using state transition diagrams. After analysing the existing system, the first step is to construct the state-space diagram. For this, an appropriate state description is required. This state description should include all the necessary information to capture the system's current state, so that if the system were interrupted and later reset to the state, the system would resume execution as if the interruption had not occured. Hence the state description would need to record which customers were at each device, how long each execution customer has been in execution, how much remaining service time is required by each executing customer, and several additional parameters that may influence the system's behaviour. But all this is too much information to be useful. Therefore, some appropriate simplification of the complete state description is needed. Thus we choose a single parameter, the number of customers currently in the system.

With this state description, it is possible to construct the appropriate state-space diagram. It is also referred to as Markov diagram. In this diagram, the system flows from state to state depending on the workload and system parameters. The state will change if either a new customer arrives or the executing customer completes service. It is assumed that at most one event, a customer arrival or a customer completion, happens at any point in time. This system is generally termed as a birth-death system.

The state of such a system can be repesented by the number of jobs (customers) n in the system. Arrival of a new job changes the state to n-1 . This is called a birth. Similarly, the departure of a job changes the system state to n-1. This is called a death. The number of jobs in such a system can therefore be modelled as a birth-death process. By solving the expressions for state probabilities, along with the expression resulting from the conservation of total probability, the model can be solved.