Invariance of the stationary distribution of queuing networks with multimode strategies and negative demands

The open queueing network with two independent simplest incoming flows of positive and negative demands with X+ and corresponding intensities is considered in article. Every demand of incoming flow of positive demands goes to l-unit independently of other demands and becomes the u-type de__N M mand, u = 1,M, with p+ _(l u) probability (££ p+ _(l u) = 1 ). Every demand of incoming flow of negative demands goes to l-unit inde- 1=1 u=1 N M pendently of other demands and becomes the u-type demand with p _0(l u) probability ( ££ p _0(l u) = 1 ). Negative u-type demand, l=1 u=1 entering in l-unit, does not require the service. It reduces the number of u-type demands in this unit by one if the queue of the unit has u-type demands, and has no effect on the state of the unit otherwise. After serving in l-unit u-type positive demand goes to k-unit instantly and independently of other demands as v-type positive demand with p++ _u)(k v) probability and as v-type negative demand with p^ _u)(k v) probability, and with p _{(l u}) ₀ probability this u-type posi- N M ___ g y £ £ p (lu ₎₍k,v) + p _a,„)(k,v) + p ₍lu)0 = I = N; , v = M ). k=1 v=1 The single line units can work in some strategies. Every strategy responds to different degree of serviceability. The dispatching rule of demands by device of l-unit is LCFS PR (l = 1N) . The network condition at moment tis characterized by vector x(t) = (x ₁(t), x ₂(t),..., x _N(t)) , where condition of /-unit at moment t is vector x _/ (t) = (x _/ (t), j (t)) = (x _/1(t), x _t2(t),..., x _t ^)(t), j (t)) , x _n (t) is the type of demand, which stands on the final position in /-unit at moment t, x/ ₂ (t) is the type of demand, which stands on the penultimate position in /-unit at moment t, x _r n/)_ ₁ (t) is the type of demand, which stands on the first position in /-unit at moment t, x _t n(l) (t) is the type of demand, which is servicing by device in /-unit at moment t, n(/) is total amount of demands in /-unit, j (t) is the number of device strategy in /-unit at moment t, / = 1, N. Servicing time of u-type demand, located in /-unit at moment t, has the exponential distribution with ц _lu parameter. The basic device strategy is strategy 0. Switching occurs only on the neighboring strategies. During switching the device from one strategy to another one the number of demands in the unit does not change. For xj conditions, where 0 < j < r , the quantity of work, which is necessary for switching of device work strategy , is random variable with arbitrary distribution function Ф^ (j, u) and expectation value (j ). The piecewise-linear Markovian process C,(t) = (x(/), )) is considered. This process is obtained by addition to x(t) continuous y = ( l 1, ji (t) t), V2,j ₂(t)( '...' V N,j _N (t) (t) ), where V _j/(t )(t) is quantity of work, which lefts to execute from moment t for switching of device work strategy to neighboring strategy in l-unit, if device works with j _l strategy. Suppose that P = {P(x)} is stationary distribution of state probabilities of x(t) process; F(x, z) are stationary distribution functions of state probabilities of piecewise-linear Markovian process Q(t) : ( ^ ) = ( ^ 1, 2, n , j _N It was proved that stationary distribution of network state probabilities is invariant in relation to functional form of distributions of work's quantities, which are necessary for switching of device work strategies.

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