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Stochastic Processes
Published in Athanasios Christou Micheas, Theory of Stochastic Objects, 2018
so that Gδy(x)=ddtpt(x,y)|t=0, as expected (recall remark 7.10). The form of the derivative at 0 suggests that given the transition function we can obtain the infinitesimal generator, but more importantly, the converse is also feasible; given an infinitesimal generator there may be a way to solve for the transition function of the Markov process. We formalize the usefulness of such derivatives in the next definition.
An approximation method for 2-chain flexible queues with preemptive priority
Published in International Journal of Production Research, 2019
Yaolei Wang, Yiming Fan, Taozeng Zhu, Ping Cao, Jingui Xie
We can characterise a continuous-time Markov chain and describe its dynamics explicitly through its infinitesimal generator. Denote with the infinitesimal generator of the Markov chain. The constant is the rate when in state that the Markovian process makes a transition into state , which measures the average number of such transitions per unit time. In each state , there are two types of transitions: one is the arrival of a customer from outside, and the other is the completion of a service. The arrival rates from outside are the same (i.e. λ). The service completion rates depend on how many type-j customers are in service, which further depend on how many servers (one or two) are available to serve the type-j customers and how many customers are in the system. In state , the service rate for type-j customers is μ when there is only one type-j customer in the system. When there are more than one type-j customers in the system, server j+1 will serve its secondary customers if there are no type j+1 customers in the system. Hence, the service rate for type-j customers is . If there is at least one type j+1 customer in the system, server j+1 will not serve the type-j customers, and the service rate for type-j customers is μ. In the following, for convenience, we let , and be the row vector with all elements being zero, except that the jth element is one. Note that is equivalent to and . Thus, we have, for , and, for , where