Minimization of unloading cost for an exponential time-sharing queueing process

A queuing process with time-sharing and readjustments is considered. A finite number m ofconflict input flows are formed in a random environment with two states. After each service a readjustmentoccurs. After a readjustment a service starts for the queue determined by virtue of aswitching function h(⋅): {0, 1, …}m  {0, 1, …, m} defined on the set of queues' lengths. Serviceand readjustment durations are exponentially distributes random variables. A mathematical modelis constructed in form of a homogeneous denumerable continuous-time Markov chain {(ƒ(t), ƒ(t),ƒ(t)); t ≥ 0} describing the evolution of the server's state, fluctuation of the queues' sizes andchanges in the state of the external random environment. A decomposition of the state space ofthe Markov chain into three non-intersecting sets is assumed, S0  ∅ being the set of admissiblestates, S  ∅ being the set of final states, and S being the set of forbidden states. Define byƒ(ƒ) = inf {t ≥ 0: (ƒ(t), ƒ(t), ƒ(t))  S+, (ƒ(t), ƒ(t), ƒ(t)) ∉ S−, 0 ≤ t ≤ t} the time to reach S withprohibition to visit S. Given the cost c(ƒ, x, k) of sojourn in a state (ƒ, x, k) per time unitJ h S S S E x kS + −ƒ = ƒ ƒ ƒ ƒ = ƒ ƒ = ƒ = ƒ<determines the average cost of reaching the final state set S from S0 without visiting the prohibitedstates of set S under the switching function h(⋅). With specific choice of S, S0, S this magnitudecan be an estimate of the load of the queuing system. In the paper systems of linear algebraicequations to compute J(h, S+, S0, S−) are obtained. Results of numerical experiments of minimizationof J(h, S+, S0, S−) in class including servicing the longest queue and threshold switching functionsare presented and discussed as well.

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