We derive two kinds of rate conservation laws for describing the time-dependent behavior of a process defined with a stationary marked point process and starting at time 0. These formulas are called TRCLs (time-dependent rate conservation laws). It is shown that TRCLs are useful to study the transient behaviors of risk and storage processes with stationary claim and supply processes and with a general premium and release rates, respectively. Detailed discussions are given for the severity for the risk process, and for the workload process of a single-server queue.

Operator methods are used in this paper to systematically analyze the behavior of the Jackson network. Here, we consider rarely treated issues such as the transient behavior, and arbitrary subnetworks of the total system. By deriving the equations that govern an arbitrary subnetwork, we can see how the mean and variance for the queue length of one node as well as the covariance for two nodes vary in time.

We can estimate the transient behavior by deriving a stochastic upper bound for the joint distribution of the network in terms of a judicious choice of independent M/M/1 queue-length processes. The bound we derive is one that cannot be derived by a sample-path ordering of the two processes. Moreover, we can stochastically bound from below the process for the total number of customers in the network by an M/M/1 system also. These results allow us to approximate the network by the known transient distribution of the M/M/1 queue. The bounds are tight asymptotically for large-time behavior when every node exceeds heavy-traffic conditions.

The queue with an infinite number of servers with a semi-Markovian arrival process and with negative exponential service times is studied. The queue length process and the type of the last customer to join the queue before time t are studied jointly, both in continuous and in discrete time. Limiting stationary probabilities are also obtained.