M/G/1/MLPS Queue Mean Delay Analysis Samuli Aalto (TKK) in cooperation with Urtzi Ayesta (LAAS-CNRS) and Eeva Nyberg-Oksanen

Outline Introduction DHR service times IMRL service times Ongoing work

Application: bandwidth sharing in IP networks Consider a bottleneck link in an IP network loaded with elastic flows, such as file transfers using TCP if RTTs are of the same magnitude, then approximately fair bandwidth sharing among the flows Intuition says that favouring short flows reduces the total number of flows, and, thus, also the mean delay at flow level (that is, the average file transfer time) How to service flows and how to analyse? Guo and Matta (2002), Feng and Misra (2003), Avrachenkov et al. (2004), Aalto et al. (2004,2005,2006)

Queueing model Assume that flows arrive according to a Poisson process with rate l flow size distribution is ”heavier” than exponential, such as hyperexponential or Pareto So, we have a M/G/1 queue at flow level customers in this queue are flows (and not packets) service time = the total number of packets to be sent attained service = the number of packets already sent remaining service = the number of packets still left

Optimality results for M/G/1 Schrage (1968) If the remaining service times are known, then SRPT is optimal minimizing the mean delay Yashkov (1987) If only the attained service times are known, then DHR implies that FB is optimal minimizing the mean delay Remark: We consider work-conserving (WC) and non-anticipating (NA) service disciplines p such as FB, MLPS, and PS (but not SRPT) for which only the attained service times are known

Service disciplines at flow level FB = Foreground-Background = LAS = Least Attained Service Choose a packet from the flow with least packets sent MLPS = Multi Level Processor Sharing Choose a packet from a flow with less packets sent than a given threshold PS = Processor Sharing Without any specific scheduling policy at packet level, the elastic flows are assumed to divide the bottleneck link bandwidth evenly Reference model: M/G/1/PS

MLPS disciplines Xi(t) a t Definition: MLPS discipline, Kleinrock, vol. 2 (1976) based on the attained service times Xi(t) N+1 levels defined by N thresholds a1 < … < aN between levels, a strict priority is applied within a level, an internal discipline (FB, PS, or FCFS) is applied Xi(t) a FCFS+PS(a) PS FCFS t

Our objective We compare MLPS disciplines in terms of the mean delay MLPS vs PS MLPS vs MLPS We assume that service times are DHR IMRL Note: DHR Ì IMRL

References K. Avrachenkov, U. Ayesta, P. Brown and E. Nyberg (2004) IEEE INFOCOM S. Aalto, U. Ayesta and E. Nyberg-Oksanen (2004) ACM SIGMETRICS – PERFORMANCE S. Aalto, U. Ayesta and E. Nyberg-Oksanen (2005) Operations Research Letters 33 S. Aalto and U. Ayesta (2006a) S. Aalto and U. Ayesta (2006b) to appear in Journal of Applied Probability S. Aalto (2006) to appear in Mathematical Methods of Operations Research

Outline Introduction DHR service times IMRL service times Ongoing work

DHR service times Service time distribution: Density function: Hazard rate: Definition: Service time distribution belongs to class DHR (Decreasing Hazard Rate) if h(x) is decreasing Examples: Pareto (taking values from 0 on) and hyperexponential

Unfinished truncated work Ux Customers with attained service Xi(t) less than x: Unfinished truncated work with truncation threshold x: Unfinished work:

Optimality of FB Xi(t) Ux(t) s s x x t t Aalto et al. (2004): FB minimizes the unfinished truncated work for any x and t in each sample path Xi(t) Ux(t) FCFS FB s s x x t t

Idea of the mean delay comparison Kleinrock (1976): For all WC & NA service disciplines p so that (by applying integration by parts) Thus,

MLPS vs PS Aalto et al. (2004): Two levels with FB and PS allowed as internal disciplines Aalto et al. (2005): Any number of levels with FB and PS allowed as internal disciplines FB FB/PS PS FB/PS FB/PS

Mean unfinished truncated work bounded Pareto service time distribution

MLPS vs MLPS: changing internal disciplines Aalto and Ayesta (2006a): Any number of levels with all internal disciplines allowed MLPS derived from MLPS’ by changing an internal discipline from PS to FB (or from FCFS to PS) MLPS MLPS’ FB/PS PS/FCFS

MLPS vs MLPS: splitting FCFS levels Aalto and Ayesta (2006a): Any number of levels with all internal disciplines allowed MLPS derived from MLPS’ by splitting any FCFS level and copying the internal discipline MLPS MLPS’ FCFS FCFS FCFS

MLPS vs MLPS: splitting PS levels Aalto and Ayesta (2006a): Any number of levels with all internal disciplines allowed The internal discipline of the lowest level is PS MLPS derived from MLPS’ by splitting the lowest level and copying the internal discipline Splitting any higher PS level is still an open problem (contrary to what we thought in an earlier phase)! MLPS MLPS’ PS PS PS

Outline Introduction DHR service times IMRL service times Ongoing work

IMRL service times Service time distribution: H-function: Mean residual lifetime: Definition: Service time distribution belongs to class IMRL (Increasing Mean Residual Lifetime) if H(x) is decreasing Examples: all DHR service time distributions, Exp+Pareto

Level-x workload Customers with attained service less than x: Unfinished truncated work with truncation threshold x: Level-x workload: Workload = unfinished work:

Non-optimality of FB Xi(t) Vx(t) s s x x t t Aalto and Ayesta (2006b): FB does not minimize the level-x workload Xi(t) Vx(t) FCFS FB not optimal FB s s x x t t

Idea of the mean delay comparison Righter et al. (1990): For all WC & NA service disciplines p so that (by applying integration by parts) Thus,

MLPS vs PS Aalto (2006): Any number of levels with FB and PS allowed as internal disciplines Aalto and Ayesta (2006b): FB/PS PS FB/PS FB/PS

bounded Pareto service time distribution Mean level-x workload bounded Pareto service time distribution

Non-optimality of FB Aalto and Ayesta (2006b): FB does not necessarily minimize the mean delay for IMRL service times Counter-example: Exp+Pareto belongs to IMRL but not DHR (for 1 < c < e): There is e > 0 such that FB FB FCFS

Outline Introduction DHR service times IMRL service times Ongoing work

Gittins index Gittins (1989): J-function Gittins index for a customer with attained service a: Gittins discipline serves the customer with highest index Gittins discipline minimizes the mean delay in M/G/1 (among NA disciplines): If DHR, then FB optimal If NBUE, then FCFS optimal If CHR+DHR, then FB, FCFS, or FCFS+FB optimal

Bandwidth sharing networks multiple links shared by elastic flows with different routes necessary stability conditions: for all links l, Verloop et al. (2005): global SRPT instable in the network case, that is necessary stability conditions are not sufficient However, local SRPT (applied to each route separately) is stable and reduces the mean delay in an optimal way

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