INPUTS

N

number of requests in the system (scalar, N > 0).

D(k)

service demand of service center $k$ (vector of length $K$, D(k) ≥ 0).

S(k)

mean service time at center $k$ (vector of length $K$, S(k) ≥ 0).

V(k)

visit ratio to center $k$ (vector of length $K$, V(k) ≥ 0).

m(k)

number of servers at center $k$. This function only supports $M/M/1$ queues, therefore m must be ones(size(S)).

Z

external delay (think time, Z ≥ 0, scalar). Default is 0.

OUTPUTS

Xl

Xu

Lower and upper bound on the system throughput. If Z>0, these bounds are computed using $Geometric\; Square-root\; Bounds$ (GSB). If Z==0, these bounds are computed using $Geometric\; Bounds$ (GB)

Rl

Ru

Lower and upper bound on the system response time. These bounds are derived from Xl and Xu using Little’s Law: Rl = N / Xu - Z, Ru = N / Xl - Z

Ql(k)

Qu(k)

lower and upper bounds of center $K$ queue length.

REFERENCES

• G. Casale, R. R. Muntz, G. Serazzi, Geometric Bounds: a Non-Iterative Analysis Technique for Closed Queueing Networks, IEEE Transactions on Computers, 57(6):780-794, June 2008. 10.1109/TC.2008.37

In this implementation we set $X^+$ and $X^-$ as the upper and lower Asymptotic Bounds as computed by the qncsab function, respectively.