Function File: [U, R, Q, X] = qncsmvald (N, S, V)

Function File: [U, R, Q, X] = qncsmvald (N, S, V, Z)

Mean Value Analysis algorithm for closed, single class queueing networks with K service centers and load-dependent service times. This function supports FCFS, LCFS-PR, PS and IS nodes. For networks with only fixed-rate centers and multiple-server nodes, the function qncsmva is more efficient.

INPUTS

N

Population size (number of requests in the system, N ≥ 0). If N == 0, this function returns U = R = Q = X = 0

S(k,n)

mean service time at center k where there are n requests, 1 ≤ n ≤ N. S(k,n) = 1 / \mu_{k}(n), where \mu_{k}(n) is the service rate of center k when there are n requests.

V(k)

average number of visits to service center k (V(k) ≥ 0).

Z

external delay ("think time", Z ≥ 0); default 0.

OUTPUTS

U(k)

utilization of service center k. The utilization is defined as the probability that service center k is not empty, that is, U_k = 1-\pi_k(0) where \pi_k(0) is the steady-state probability that there are 0 jobs at service center k.

R(k)

response time on service center k.

Q(k)

average number of requests in service center k.

X(k)

throughput of service center k.

NOTES

In presence of load-dependent servers, the MVA algorithm is known to be numerically unstable. Generally this problem manifests itself as negative response times or utilization.

REFERENCES

• M. Reiser and S. S. Lavenberg, Mean-Value Analysis of Closed Multichain Queuing Networks, Journal of the ACM, vol. 27, n. 2, April 1980, pp. 313–322. 10.1145/322186.322195

This implementation is described in G. Bolch, S. Greiner, H. de Meer and K. Trivedi, Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications, Wiley, 1998, Section 8.2.4.1, “Networks with Load-Dependent Service: Closed Networks”.

See also: qncsmva.

Package: queueing