Function Reference: qnom

Function File: [U, R, Q, X] = qnom (lambda, S, V)

Function File: [U, R, Q, X] = qnom (lambda, S, V, m)

Function File: [U, R, Q, X] = qnom (lambda, S, P)

Function File: [U, R, Q, X] = qnom (lambda, S, P, m)

Exact analysis of open, multiple-class BCMP networks. The network can be made of single-server queueing centers (FCFS, LCFS-PR or PS) or delay centers (IS). This function assumes a network with K service centers and C customer classes.

INPUTS

lambda(c): If this function is invoked as qnom(lambda, S, V, …), then lambda(c) is the external arrival rate of class c customers (lambda(c) ≥ 0). If this function is invoked as qnom(lambda, S, P, …), then lambda(c,k) is the external arrival rate of class c customers at center k (lambda(c,k) ≥ 0).
S(c,k): mean service time of class c customers on the service center k (S(c,k)>0). For FCFS nodes, mean service times must be class-independent.
V(c,k): visit ratio of class c customers to service center k (V(c,k) ≥ 0 ). If you pass this argument, class switching is not allowed
P(r,i,s,j): probability that a class r job completing service at center i is routed to center j as a class s job. If you pass argument P, class switching is allowed; however, all servers must be fixed-rate or infinite-server nodes (m(k) ≤ 1 for all k).
m(k): number of servers at center k. If m(k) < 1, enter k is a delay center (IS); otherwise it is a regular queueing center with m(k) servers. Default is m(k) = 1 for all k.

OUTPUTS

U(c,k): If k is a queueing center, then U(c,k) is the class c utilization of center k. If k is an IS node, then U(c,k) is the class c traffic intensity defined as X(c,k)*S(c,k).
R(c,k): class c response time at center k. The system response time for class c requests can be computed as dot(R, V, 2).
Q(c,k): average number of class c requests at center k. The average number of class c requests in the system Qc can be computed as Qc = sum(Q, 2)
X(c,k): class c throughput at center k.

NOTES

If the function call specifies the visit ratios V, class switching is not allowed. If the function call specifies the routing probability matrix P, then class switching is allowed; however, all nodes are restricted to be fixed rate servers or delay centers: multiple-server and general load-dependent centers are not supported. Note that the meaning of parameter lambda is different from one case to the other (see below).

REFERENCES

Edward D. Lazowska, John Zahorjan, G. Scott Graham, and Kenneth C. Sevcik, Quantitative System Performance: Computer System Analysis Using Queueing Network Models, Prentice Hall, 1984. http://www.cs.washington.edu/homes/lazowska/qsp/. In particular, see section 7.4.1 ("Open Model Solution Techniques").

See also: qnopen,qnos,qnomvisits.

Demonstration 1

The following code

 P = zeros(2,2,2,2);
 lambda = zeros(2,2);
 S = zeros(2,2);
 P(1,1,2,1) = P(1,2,2,1) = 0.2;
 P(1,1,2,2) = P(2,2,2,2) = 0.8;
 S(1,1) = S(1,2) = 0.1;
 S(2,1) = S(2,2) = 0.05;
 rr = 1:100;
 Xk = zeros(2,length(rr));
 for r=rr
   lambda(1,1) = lambda(1,2) = 1/r;
   [U R Q X] = qnom(lambda,S,P);
   Xk(:,r) = sum(X,1)';
 endfor
 plot(rr,Xk(1,:),";Server 1;","linewidth",2, ...
      rr,Xk(2,:),";Server 2;","linewidth",2);
 legend("boxoff");
 xlabel("Class 1 interarrival time");
 ylabel("Throughput");

Produces the following figure

Figure 1

Package: queueing