More specifically, networks which can me analyzed by this function have the following properties:

• There exists only a single class of customers.
• The network has $K$ service centers. Center $k\; \backslash in\; 1,\; \dots ,\; K$ has $m\_k\; >\; 0$ servers, and has a total (finite) capacity of $C\_k\; \backslash geq\; m\_k$ which includes both buffer space and servers. The buffer space at service center $k$ is therefore $C\_k\; -\; m\_k$.
• The network can be open, with external arrival rate to center $k$ equal to $\backslash lambda\_k$, or closed with fixed population size $N$. For closed networks, the population size $N$ must be strictly less than the network capacity: .
• Average service times are load-independent.
• $P\_i,\; j$ is the probability that requests completing execution at center $i$ are transferred to center $j$, $i\; \backslash neq\; j$. For open networks, a request may leave the system from any node $i$ with probability $1-\backslash sum\_j=1^K\; P\_i,\; j$.
• Blocking type is Repetitive-Service (RS). Service center $j$ is $saturated$ if the number of requests is equal to its capacity $C\_j$. Under the RS blocking discipline, a request completing service at center $i$ which is being transferred to a saturated server $j$ is put back at the end of the queue of $i$ and will receive service again. Center $i$ then processes the next request in queue. External arrivals to a saturated servers are dropped.

INPUTS

lambda(k)

N

If the first argument is a vector lambda, it is considered to be the external arrival rate lambda(k) ≥ 0 to service center $k$ of an open network. If the first argument is a scalar, it is considered as the population size N of a closed network; in this case N must be strictly less than the network capacity: N < sum(C).

S(k)

average service time at service center $k$

C(k)

capacity of service center $k$. The capacity includes both the buffer and server space m(k). Thus the buffer space is C(k)-m(k).

P(i,j)

transition probability from service center $i$ to service center $j$.

m(k)

number of servers at service center $k$. Note that m(k) ≥ C(k) for each k. If m is omitted, all service centers are assumed to have a single server (m(k) = 1 for all $k$).

OUTPUTS

U(k)

center $k$ utilization.

R(k)

response time on service center $k$.

Q(k)

average number of customers in the service center $k$, $including$ the request in service.

X(k)

throughput of service center $k$.

NOTES

The space complexity of this implementation is $O(\backslash prod\_k=1^K\; (C\_k\; +\; 1)^2)$. The time complexity is dominated by the time needed to solve a linear system with $\backslash prod\_k=1^K\; (C\_k\; +\; 1)$ unknowns.