Queues in Multichannel Systems Remotely Controlled Via a Common Communication Link

Abstract

In some control problems where human decision constitutes part of a feedback loop and the decision maker is remote from the plants, the corrective error signals are transmitted in the form of messages via a single communication link. Clearly, a queue of messages may be formed and the problem arises as to how rapidly the system outputs can be corrected. Consider the case of a number of independent plants so remotely controlled. Each of the n inputs constitutes a channel. An automatic sorting mechanism which connects the communication link to the inputs of the plants receives and channels the messages. This paper considers two cases: 1) The sorting mechanism examines only the first message in line, causing all messages behind to wait if the specific channel required by the first message is busy, even if another message in line is addressed to an idle channel. By "busy" it is meant that the output of the specific plant is being corrected as a result of a previous message. This case is novel. 2) The sorting mechanism searches the queue until it finds a message addressed to a channel that is presently idle. Under the assumptions of exponentially distributed service time, negligible sorting time, and Poisson distributed output errors, this paper uses a generating function approach and presents the queue size and waiting time distributions to provide statistical information concerning the time lag in the feedback loop.