COMPARISON OF HYBRID MINIMUM LAXITY 1ST-IN-1ST-OUT SCHEDULING POLICIES FOR REAL-TIME MULTIPROCESSORS

In this paper we study the behavior of two policies for scheduling customers with deadlines until the beginning of service onto multiprocessors. Both policies attempt to approximate the performance of the minimum laxity scheduling policy without incurring its complete overhead. This is accomplished by dividing the queue into two queues-one, of maximum size n > 0, managed using the minimum laxity policy and another of unbounded size managed in a first in first out manner. One policy, F/ML(n), places the ML queue at the front, i.e., customers finding n or more in the system enter the First-In-First-Out (FIFO) queue which in turn feeds the ML queue. The other policy, ML(n)/F, places the ML queue at the back, i.e, arriving customers enter the ML queue and if the total number in the system exceeds n, forces one customer from the ML queue to the FIFO queue. We show that these seemingly dissimilar policies exhibit exactly the same behavior for a fixed value of n both when customers are allowed to be discarded when they miss their deadlines before entering service and when they are not allowed to be discarded. We also establish monotonicity properties for both policies. In the case that no customer is discarded, we show that the expectation of a convex function of the stationary customer tardiness, the difference between the departure time and deadline, is a decreasing function of n. In the case that discards are allowed, we show that the number of customers lost within an interval of time is a decreasing function of n in the sense of stochastic ordering. Last, all of the results, except for the monotonicity result for systems with discard are shown to hold when deadlines are until the end of service.