ON A DUALITY PRINCIPLE IN PROCESSES OF SERVICING MACHINES WITH DOUBLE CONTROL*

This paper examines a doubly controlled process of servicing machines. The classical system treated by Tals is equipped with m + 1 unreliable machines served by one repairman. In the present medification of this model, the failure rates and the repair time may be controlled with respect to the state of the system. The process describing the number of intact machines is considered. To derive its steady state distribution in the form of a simple explicit formula, the author introduces an auxiliary model with m unreliable machines and a single repairman who keeps working even when all machines are intact. This result is based on a duality principle applied to the process above.


I. INTRODUCTION
In [2] the author studied a multi-channel loss (transportation) queueing system with control of input stream and service. The servicing facility of the system contained m parallel channels processing a stream of singly arriving customers. No customer was accepted when the servicing facility was occupied. The input stream and service wcrc subject to a comprehensive control.
This model can also be interpreted as a system of m unreliable machines scrvcd by a single repairman with corresponding service and failure rates control. More specifically, each of the working machines can break down with a rate dependent on the total number of intact machines. The repair time also depends upon the number of intact machines. The repairman is not idle even when all machines become intact. At these times (called regenerations of the system), the repairman leaves the system and comes back later with a new machine that immediately replaces an available defective machine. If no machine breaks down during the repairman's absence, a substitution (of one of the working machines) takes place at the repairman's own choice, but the number of working machines does not change (it is still m), and this action is supposed not to affect the future status of the system. Since the repairman leaves the system to get new equipment, we can restrict his absence to a definite time. So the control can particularly b applied to this situation. Besides the clear advantages of this system as a doubly controlled wansportation servicing model, it can also be viewed as a doubly controlled model of servicing machines. Another more popular model of servicing machines with idle periods is one with m + 1 machines and with a single repairman who is either repairing defective machines or simply keeping watch when all m + 1 are intact. However, he is alert to start his service as soon as any working machine fats. Such a system is probably more practical than the one described above although obviously with a more difficult analysis. Without control it was originally set and studied by Takics [3] in his fundamental work. Takeics solved many problems related to this system.
In the present paper, the author studies the process of servicing machines for the model with idle periods. The servic time and failure rams are subject to control at moments of time appropriate for decision making. The process describing the number of intact machines is treated. By using sonm semi-regenerative techniques and by introducing a duality principle, the author obtains an explicit relation between the two models. The results are combined with those derived in [2].

MODEL DESCRIPTION
The models introduced in section I arc formalized as follows. Model I. The system consists of m + 1 unreliable machines served by a single repairman. Denote by Z I the number of intact machines at time > 0. When the repair of a last defective machine is completed and the total number of intact machines becomes m + I, the repairman is idle until the next breakdown. Let "1' *2,'" be the successive instants of the completion of machine repairs. The repair time of the nth machine is distributed in accordance with A ,(x)e {Ai(x);/c 0, 1, m} (a tuple of arbitrary d.f.'s), where ," Z *,t-' n 1, 2, That is, the repair time depends upon the number of intact machines at the moment immediately before the completion of the preceding service. The working machines perform certain jobs. Within the interval [n, n+1) the continuous On a Duality Principle in Processes of Servicing Machines with Double Control: Dshalalow 247 durations of each job are conditionally independent given a and exponentially distributed with parameter p {/a,, k--O, 1, ...,m + w} C: R+\{0}. MODEL 2. There arc a maximum of m working unreliable machines and one repairman. The total number of intact machines at time t 0 is denoted by Z c Unlike Model I, there arc no idle periods. At certain epochs of time the repairman temporarily abandons the system to acquire a new machine. Let Ta be an epoch when a machine is completely repaired. If at this time the total number of intact machines is m, the repairman leaves the system and returns to the system at time Ta+I with a new machine to replace any machine that has failed during the repairman's absence. In other words, if the repairman returns to fewer than m working machines, then the number of working machines increases by one. If no machine has failed to this time, a replacement still occurs but at the repairman's own choice (in this case, without any effect on the system). In both cascs used machines arc removed from the system. Consequently, T I, T 2, arc the moments when the repairman completes a job (repair of a machine or acquisition of new equipment).
At time Ta he begins with the repair of a current machine or leaves the system if no defective machine is available. At the time Ta+1 he completes the repair or returns to the system with a new machine. The length of the interval [Ta, Ta+ I) is distributed according to the d.f. As mentioned, this model is identical to the doubly controlled m-channel loss qucucing system studied by the author [2], where Z denotes the number of customers in the system at time t. To explain equivalence between both systems, we use the mutual notation below.
At time T n a customer departs from a source and at time Tn+ arrives at the system. The customer is served by one of the free parallel channels available, or is lost by the system if the servicing facility is busy. The length of the interval IT n, Tn+ ) is dislributed as in (2.1) and (2.2), and the servicing policy is determined by (2.3). Within the interval [Tn' Tn+l ) the service durations of customers in each of the channels are conditionally independent given X n and exponentially distributed with parameter/.t X In addition to the above formalism, we use the following convention. Throughout the paper, we drop the superscripts in all functionals of A(x) and of/z if the corresponding indices in A(x) and/ are missing or if brevity dictates as in the cases of Am(x) and lam.