Asymptotic analysis of resource infinite-server queueing tandem with MMPP arrivals

In this work, the MMPP/(GI/«)^M queueing system with unlimited number of servers and with unlimited amount of resource is studied. The arrival process is a MMPP. Service times on each server on m^th phase are i.i.d. with distribution function Bm(x) (m = 1M). All customers form request for a random resource v with distribution function G(y) = P {v < y} on each phase. After the service end on M^th phase the customer leaves system and sets free the occupied resource. Consider 2M-dimensional stochastic process {i(t), v(t)} = {j(t),...,г_м(t),V(t),.. V_M(t)} , where im(t) and Vm(t) denote the numbers of customers and the total resources at the m^th phase in the system at time t, respectively. We proposed the dynamic screening method for its investigation. Note that this method exactly determines the characteristics of the process V(t) since the screened process contains only those customers, which do not finish the service at the moment T. The system of Kolmogorov differential equations is derived. By using the partial characteristic function, we obtained the main equation: SH(u, v,t) , ч -= H (u, v,t) St ^V ' Л Mm )[e^JUmG* (v_m)-1l + Q m=1 with the initial condition H(u v to ) = r . To solve this equation, the method of asymptotic analysis is proposed under the condition of an infinitely growing arrival rate. We proved that in steady state regime the characteristic function of the customers numbers and the total resources at the system phases corresponds to a 2M-dimensional Gaussian distribution with parameters: · expectations vector Y = m[Yj Y₂ ... Y_L], where Y =[1 ^ЩВ-Дt)-B*(t))dT , matrix W = N(xW¹) + kW*²⁾) , where - covanance _K(2) _K(2) ^K1 ^K12 K²) T^(²) _K 2) ^K1 L _K 2) ^K2L K₁⁽ (1) (1) 0K (1) <2) W W 0 0 ... kL¹) Г(2) 1^(2) K_L2) 1 a 1 a ^ a j² да , , J(B*-1 (x)_ B* (x))dx , k(²): K (¹⁾ = J (b*-1 (x)- B* (x)) dx , j (b-x)_ b*( x))( b*-x)-b*( x)) dx . 1 «1 k( ²) = ^Kj

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