Stability of size-based scheduling in resource-sharing networks Maaike Verloop CWI & Utrecht U. Sem Borst CWI & Eindhoven U.T. & Lucent Bell Labs Sindo Núñez-Queija CWI & Eindhoven U.T

2 Introduction Size-based scheduling in single resource systems SRPT, LAS, … Data flows: simultaneous resource possession Not work conserving Performance [Yang & De Veciana] Performance measures –Stability –Delay –Resource occupancy Compare re-entrant lines and interacting dynamical systems server users queue

3 Introduction Size-based scheduling in single resource systems SRPT, LAS, … Data flows: simultaneous resource possession Not work conserving Performance [Yang & De Veciana] Performance measures –Stability (not trivial) –Delay –Resource occupancy Compare re-entrant lines and interacting dynamical systems 1 2 0

4 Outline Model description Stability of size-based scheduling –SERPT: Shortest Expected Remaining Processing Time –SRPT: Shortest Remaining Processing Time –LAS: Least Attained Service

5 Model description Linear network L nodes, with capacity 1 L +1 classes of users Poisson arrival processes with rate λ i Random flow size B i with mean β i Traffic load ρ i = λ i β i N i denotes the number of class-i flows in the system class 0 class 2class 3class 1 class L

6 Stability Class i is stable iff P( N i =0) > 0 Network is stable if all classes are stable Necessary condition for stability of network: ρ 0 +ρ i < 1 for all i Sufficient condition (no parallelism): ρ 0 +ρ 1 +…+ ρ L < 1 for all i 123L 0

7 Stability conditions depend on disciplines Prioritize class 0 –Class i is served only if class 0 is empty –Stable iff ρ 0 +ρ i <1, for all nodes 123L 0 standard conditions

8 Stability conditions depend on disciplines Prioritize class 0 –Class i is served only if class 0 is empty –Stable iff ρ 0 +ρ i <1, for all nodes Prioritize all classes 1,…,L –Class 0 is served only if classes 1,…,L are empty –Stable iff –More stringent stability condition 123L 0 standard conditions

9 Size-based scheduling I: SRPT Class 0 is served at full rate if a class-0 user has the shortest remaining size among all users Otherwise, at each node i, class i is served at full rate If N i > 0, node i works at full capacity, –Class i is stable iff ρ 0 +ρ i < 1 Stability condition for class 0 –Largest flows that get through –ρ 0 (x 0 ) + ρ i (x i ) ≤ 1 –x 0 ≤ x i 123L 0

10 SRPT: Stability of class 0 Time-scale decomposition: large class-0 flows –Arrival rate: λ 0 (ε)= ελ 0 –Service requirements: B 0 (ε)=B 0 /ε –Traffic load independent of ε: ρ 0 (ε)= ελ 0 β 0 /ε =ρ 0 Distinguish between class-i flows that are larger or smaller than 1/√ε –Calculate P(no i-flow is smaller than 1/√ε) Class 0 is stable in the ε-system for ε small enough

11 Short class-0 flows Assume that class-0 flows are shorter than those of all other classes: M 0 < m i (almost strict prioritization) Then class 0 is stable under standard conditions: ρ 0 +ρ i <1 SRPT: Stability of class 0 (cont.)

12 Size-based scheduling II: LAS In each node a flow has the right to a share of the capacity if it is one of the shortest Class-0 flows can only utilize the smallest share along the route Surplus capacity is re-allocated to the other classes  if N i > 0, node i works at full capacity Class i is stable iff ρ 0 +ρ i < 1 123L 0

13 LAS: Stability of class 0 ε-system: relatively large class-0 users –Arrival rate: λ 0 (ε)= ελ 0 –Service requirements: B 0 (ε)=B 0 /ε –Load independent of ε: ρ 0 (ε)= ρ 0 –Distinguish between “long” and “short” flows Class 0 is stable in the ε-system for ε small enough

14 class 0 class 1 N1N1 N0N0 N 2 =0 Conclusion Size-based schedulers may render poor performance in networks Study performance of schemes such as α-fair allocations that are known to ensure stability Optimal allocation schemes needed to provide a sensible benchmark –Complexity / approximations –Linear network –More general networks

Stability of size-based scheduling in resource-sharing networks Maaike Verloop Sem Borst Sindo Núñez-Queija