Synchronous Rate and Frequency Adjustment for Utilization Control in Distributed Real-Time Embedded Systems

Ti, i = 1, . . . ,m the ith end-to-end task in the system Tij , j = 1, . . . , ti the jth subtask of Ti cij the estimated execution time of Tij when the processor holding Tij runs at the highest processor frequency ri the task rate of Ti and its subtasks Tij , j = 1, . . . , ti Ri = {γi1, . . . , γil} the predefined set of ri Pq , q = 1, . . . , n the qth processor in the system Sq the set of subtasks Tij held by Pq fq the normalized CPU frequency of Pq [fmin, 1] the range for fq uq CPU utilization of Pq uq CPU utilization setpoint of Pq gq workload variation factor G = diag(g1, . . . , gn) ∈ Rn×n load-variation matrix F = diag( 1 f1 , . . . , 1 fn ) ∈ Rn×n frequency-inverse matrix E = (eij) ∈ Rm×n subtask allocation matrix r = [r1, . . . , rm] ∈ Rm task rate vector u = [u1, . . . , un] ∈ Rn processor utilization vector v ∈ Rn utilization measurement noise Ts the length of sampling period