Research and optimization of semi-Markov queueing models

Investigation of Queuing Systems in System Structure Management

“…In this section we repeat some definitions presented [22][23][24] and then derive some relations for the semi-Markov kernel and probability measures used in construction of the accumulation functional. The classic case [22] has a semi-Markov process that is 𝜉(𝑡) defined by a homogeneous two-dimensional Markov chain or a homogeneous Markov renewal process:…”

“…. Given the numerical functions 𝑅 𝑖𝑗 (𝑡, 𝑢), 𝑖, 𝑗 ∈ 𝐸, the value of the functional corresponding to the trajectory defined above is determined by ∑ 𝑅 𝑖 𝑘 𝑖 𝑘+1 (𝑡 𝑘+1 , 𝑢 𝑘+1 ) + 𝑅 𝑖 𝑛(𝑇) 𝑖 𝑛(𝑇)+1 𝑛(𝑇)−1 𝑘=0 (22) The function 𝑅 𝑖𝑗 (𝑡, 𝜏, 𝑢) is the mathematical expectation of the accumulated effect (income) over time 𝜏, provided that the process is in state 𝑖 and the next state will be state 𝑗, and the time of this transition is 𝑡, provided that the decision 𝑢 is made at the time of the transition from state 𝑖.…”

“…Consequently, the specific accumulated effect is taken as a characteristic of the functioning quality, that is, 𝑙𝑖𝑚 𝑡→∞ 𝑆 𝑖 (𝑡) 𝑡 (where this limit exists). According to the accumulation functional theorem [22], if there is a semi-Markov process with a finite set of states, the nested Markov chain is ergodic, at least one of the distributions 𝑄 𝑖𝑗 (𝑡) is indecisive, and the average revenues 𝑠 𝑖 are finite for any 𝑖 ∈ 𝐸, then it can be defined as follows:…”

Optimization model of transport traffic at preliminary registration of queries based on controlled queueing models

“…Note that it is very reasonable to change the characteristics of arrival flows in various models for optimization of its functioning. The control is based on the theory of controlled semi-markov processes for system optimization [10][11][12]. The control is carried out using a type of the next batch, the moments of batch arrivals and the quantity of queries in the batch.…”

“…Note that, CBSMAP-flow is also good for modeling of data-flows and other flows. All major characteristics for the CBSMAP-model was obtained in the previous papers [10][11][12].…”