Longest path analysis in networks of queues: Dynamic scheduling problems

In this paper, we develop an acyclic network of queues for the design of a dynamic flow shop, where each service station represents a machine. The orders arrive at the input station according to a Poisson process, assuming each order consists of n jobs. Jobs associated with successive orders contend for resources on a FCFS (First Come First Served) dispatching rule in all machines. Upon arriving each order, it is splitted into n jobs, in which each job joins the related queue for the first machine. Each directed path of the network of queues represents a sequence of the required operations on a particular job. The processing times are assumed to be either exponentially distributed or exhibit generalized Erlang distributions in heavy traffic condition. The transport times between every pair of service stations are also independent random variables with exponential distributions. We develop a method for approximating the distribution function of the longest path length in the network of queues by constructing a proper continuous-time Markov chain. Therefore, our method leads to determining the flow time distribution of orders, which is one important performance measure in scheduling problems. The proposed methodology is also applied to obtain the project completion time distribution in dynamic PERT networks. The results are verified by simulation.