Queuing when Jobs Require Several Services which Need Not be Sequenced

Several queuing models are considered in which a job requires a number of different services. The services are performed in any sequence, possibly simultaneously, by a number of servers. Steady state probabilities are found for two service jobs processed by two server systems. In the first system, the two servers work on the same job if they are working. In the second system, the servers may work on different jobs. If server one is working on job k, server two may be working on job k-1, k, or k + 1 but not on k-2 or k + 2. In these systems, the servers are specialists. In the third system, the servers may perform either of the two required services. If one has bogged down, the other server does both parts of each job.