api.mapqn

mapqn_amva(mu, D0s, D1s, N)

[X, QQ, U, PIK, ES] = MAPQN_AMVA(MU, D0S, D1S, N)

Horizontal-cut mean value analysis of a closed multiclass network made of an exponential infinite-server station (think rate MU(r) for class r) and one FCFS single-server station whose class-r service process is the MAP (D0S{r}, D1S{r}). The MAP of class r moves only while a class-r job is in service and is frozen otherwise, the convention of SolverCTMC.

The recursion walks the population lattice n <= N in lexicographic order and at every point solves ONE linear R x R system. Its unknowns are the per-phase means Q_r^k = E[n_r 1{k}] over the joint phase k = (k_1..k_R), the busy laws U_r^k = P[serving r, k], the phase law pi_k and the throughputs X_r. Exact relations: the joint phase balance, the class marginals U_r = X_r E[S_r] theta_r, and the per-class horizontal cut (generator balance of n_r 1{k}) of Casale-Smirni, DSN 2009. Closures: the product busy law theta_r(k_r) times the post-completion laws of the frozen MAPs (which solves the phase balance identically), the service-age closure of the cross term E[n_r 1{serving s}] (class r accumulates at its throughput over the elapsed class-s service, whose mean given the phase is theta_s(-D0_s)^{-1} / theta_s), and Little’s law resolved by arrival phase with the exact FCFS response of the queue composition seen at population n - e_r (the multiclass arrival theorem). K_r = 1 for every class reproduces multiclass FCFS MVA on class means.

Inputs:: mu - 1 x R think rates (exponential infinite server) D0s - 1 x R cell, D0s{r} the K_r x K_r hidden-transition matrix of class r D1s - 1 x R cell, D1s{r} the K_r x K_r completion matrix of class r N - 1 x R populations (a class with N(r) = 0 is absent)
Outputs:: X - 1 x R class throughputs Qq - 1 x R mean queue lengths at the MAP station (in service included) U - 1 x R busy probabilities of the server per class, X(r) E[S_r] pik - 1 x prod(K_r) joint phase law at population N (class R fastest) ES - 1 x R mean service times
Reference:: G. Casale, E. Smirni, “MAP-AMVA: Approximate Mean Value Analysis of Bursty Systems”, IEEE/IFIP DSN 2009, pp. 409-418 (the horizontal cut).