How Networks of Queues Came About

T essay, about work done almost a half century ago, is written as a snippet of autobiography rather than as a technical review of the mathematics of networks of queues. To summarize, interest in jobshop manufacturing led me to a good mathematical subject area. I tried the only technique I knew that seemed applicable to one of the more obvious problems in this area; surprisingly, this technique worked, so the first findings on “networks of queues” were uncovered. Since then, original context of the results has been broadened to include a large range of systems of potential interest to workers in operations research. Now I will expand the paragraph above, filling in details, which may be interesting to some readers and which perhaps belong in the recorded history of operations research. I received my UCLA Ph.D. in very pure mathematics in 1952. The job market then pulled (or pushed) me into the early days of operations research—simply because the only interesting job I could find was with the Naval Logistics Research Project at UCLA (later renamed the Management Science Research Project). The idea of OR immediately appealed to me, and in accordance with the stated mission of the project, I was soon attracted to the problems of jobshop scheduling, epitomized by many small machine shops in the Los Angeles area. First, I carried out some primitive simulations—by hand, and later, with Bill Marlowe’s help, using the primitive wireboard-programmed logistics computer at George Washington University, and still later, with Yoshiro Kuratani’s help, using UCLA’s pioneering Bureau of Standards Western Automatic Computer (SWAC). In spite of impressively skillful programming by Marlowe and Kuratani, these machines simply were not adequate for such simulations. In retrospect, I doubt if my early ideas would have led to anything worthwhile even on the most powerful machines of today. An audited course given by Alex Mood, using William Feller’s classic 1950 textbook, Probability Theory and Its Applications, got me into elementary queueing theory and the classical models of simple Markovian waiting-line models. Perhaps because I was still so pure (mathematically), it was some time before I conceived of a slick but enormously idealized and simplified model of a machine shop, which I called a “network of queues.” In this model, all decisions were made according to nice Markovian rules: A “job” entered the “shop” at a randomly determined “machine group,” and joined a queue. When the work required there on the job was done, then with a random choice from a probability distribution attached to the machine group, the job was either (i) completed, with a certain probability, or (ii) sent on to some randomly determined machine group. Thus, all jobs that had been served at any one center, regardless of how long they had been in the shop or how many operations had been performed on them, had the same probabilities for their future courses. And, of course, each machine group was a classical queuing center, with a specified number of servers (“machines”), and an exponential probability distribution of time required for service to any job. The “state of the system” at any time was defined as the vector of components, one for each machine group, specifying the number of jobs at the center—queued or being served. I wanted to find the long-term “steady state” for the system—that is, their equilibrium probabilities for the state of the system. It should not have been really difficult to write equations for these long-run distributions of the state of the system, although it took me a while to do so. Then I found myself clueless as to how these equations might be solved, and I puzzled over this for awhile. Somehow, the idea occurred to me that it might help to pretend that their solution could be approached by supposing that each center behaved, in the long run, as if it were independent of the others, and with the long-term distribution of states got from classical queueing theory for a center with the same load of customers (“jobs”), number of servers, and distribution of time required to process a customer. Although I had trouble with notation, it was not a difficult matter to calculate the resulting distribution and grinding it through the equations. I was most pleasantly surprised to find that my “pretend solution” actually worked—that so far as long-run “steady state” or “equilibrium” probabilities are concerned, my network system behaved as if the machine groups were in fact independent classical queueing systems. This result was the main finding of my paper, “Networks of Waiting Lines” (Operations Research 5(4) 1957). A few