When customers in a queuing network model exhibit different routing patterns, different service time requirements, and/or different scheduling disciplines, the model is said to have multiple classes of customers. Consider a local area network with P client workstations and Q servers. Each client workstation may have its own local storage, but it may also require additional storage capacity from the servers. After executing at a workstation, a process sends a request across the network to acquire service from one of the storage servers. Each client workstation process places one request at a time on the network. A queuing model of this system is proposed in [22]. If all client processes have identical workload requirements and balance their service requests across all file servers, the system can be modeled by a singleclass workload with P processes. However, if each client process restricts its requests to a subset of servers, with each client visiting a different set of servers, performance can be significantly improved. By judiciously segregating clients on the servers, potential queuing delays are reduced and average response time is likewise reduced. In this alternative model, each client process has a different routing probability (i.e., each process visits a different set of servers) and a separate workload class is required to represent each client workstation. Thus the system is modeled with P classes, each with exactly one process in each class. In general, the queuing networks considered here consist of K devices (or service centers) and R different classes of customers. A central concept in the analysis and solution of queuing networks is the state of the network. The state represents a distribution of customers over classes and devices. The network state is denoted by a vector , where component (i = 1, ..., K) is a vector that represents the number of customers of each class at server i. That is, = (n_{i,1}, n_{i,2},....,n_{i,R}) where n_{i,r} represents the number of class r customers at server i. For instance, returning to the example in Figure 13.1 the state of the system is defined as (), where is a tuple specifying the number of update and query transactions at device i. Two possible states in this example include ((1,3),(0,0),(0,0)) and ((0,1),(0,2),(1,0)). The former state indicates that all transactions (i.e., one update and three query transactions) are at the processor. The latter state represents the situation where one query transaction is executing at the processor, two queries are at disk 1, and the update transaction is being serviced at disk 2. The BCMP theorem [3], developed by Baskett, Chandy, Muntz, and Palacios, specifies a combination of service time distributions and scheduling disciplines that yield multiclass productform queuing networks that lend themselves to efficient model solution techniques. Open, closed, or mixed networks are allowed. A closed network consists of only closed classes with a constant number of customers in each class at all times. In contrast, an open network allows customers in each class to enter or leave the network. A mixed network is closed with respect to some classes and open with respect to other classes. Basically, the set of assumptions required by the BCMP theorem for a productform solution is as follows.
In open networks, the time between successive arrivals is assumed to be exponentially distributed. Bulk arrivals are not allowed. Multiclass productform networks have efficient computational algorithms for their solution. The two major algorithms are convolution [6] and MVA [16]. Because of their simplicity and intuitive appeal, this chapter presents MVAbased algorithms for exact and approximate solutions of models with multiple classes. (Also see [13] for MVAbased algorithms of QNs and [8] for solution algorithms based on decomposition and aggregation techniques.) The following notation is used for the multiclass models presented here.
