The maintenance of bidirectionally patrolled machines

The problem considered is that of maintaining a set of N stations (machines, production facilities) which are set out along a line and numbered, say, from left to right, 1 to N. The stations (which are not necessarily identical) are maintained and repaired if necessary by one operative, who patrols them first from left to right in the order 1 to N and then from right to left in the order N — 1 down to 1 and so on. It is assumed that breakdowns at station i occur completely at random in running time at an average rate A,. The time for the operative to travel from left to right from station i — 1 to station i (or from right to left from station i to station i — 1) and then to carry out routine maintenance at station i is assumed to be a constant for this pair of stations, and is denoted by w,. If, on arrival at station i, the operative finds the station out of action, then an additional time r, is needed to repair station i. It is assumed that r, is a constant for station i. It is also assumed that a repair attempt at station i is successful with probability CT, (not necessarily 1). Thus the model caters for a heterogeneous set of stations, unequally spaced. Important performance measures for the system include the average time to traverse the line of stations, along with the mean availability. For individual stations, the availability, the mean time spent waiting for attention, and the mean length of the stopped period are all important. It is shown how all of these quantities can be computed. 1