Queueing Models for P2P Systems

 Extend classical queuing theory for P2P systems.  Develop taxonomy for different variations of these queuing models.  Derive stability conditions for these models. Contents

 Jobs and service stations both arrive randomly,possibly correlated.  Following questions can be answered : 1.Necessary and sufficient conditions for stability 2.Performance of given workload and service parameters Introduction

Notations

 A/B/C/E/POLICY  M/M/(M/M)/FCFS  M/M/(M/M)/PS(k)  M/M/(-/-)  M/M/(-/+G)

Stability A p2p system is stable if its corresponding job- server process {n c (t),n s (t)} is positive recurrent Transition rates for M/M/(M/M)/FCFS systems ( n c, n s ) to ( n c + 1, n s ) = ƛ c ( n c, n s ) to ( n c - 1, n s ) = n s µ c ( n c, n s ) to ( n c, n s + 1 ) = ƛ s ( n c, n s ) to ( n c, n s - 1 ) = n s µ c

Quasi Birth Death Process  Quasi birth death process is CTMC that satisfies that : 1.It has a two dimensional state space union of l(n) where l(n) is called level. 2.A transition from (n1,i) to (n2,j) is not possible if | n1- n2| >= 2  It is called homogeneous if transition rates are independent of n

M/M/(M/M)/FCFS system is homogeneous QBD process. Applying results from QBD process, stability condition for M/M/(M/M)/FCFS is ρ c < ρ v

Stability for M/M/(M/M)/PS(k) system M/M/(M/M)/PS(k) system is nonhomogeneous QBD It is stable if and only if ρ c < ρ v

 M/M/(-/-) system : It is one dimensional birth-death process exactly similar to M/M/∞ system which is always stable  M/M/(-/+G) system : Number of servers > Number of jobs, hence always stable.  As long as every peer brings in some service capability, system is always stable

Modelling P2P storage system : Wuala In Wuala,there are storage peers and downloading peers. File is divided into fragments and fragments are stored at different storage peers. Question : What type of online behavior of storage peers is necessary to satisfy download request rates ? Notations : t on – average online time of a storage peer t off – average offline time of a storage peer r u – upload bandwidth of each storage peer r d – download bandwidth of downloading peer L – average file length

Assumptions : Server ON-OFF times and file lengths are exponentially distributed. Arrival of file download requests to follow a Poisson process then such a P2P system can be modeled by an M/M/(M/M)/PS(k) model Here ƛ s = 1/t off, µ s = 1/t on ƛ = ɣ, µ c = r u / L and k = r d / r u

Stability condition reduces to

Discussion and Future Work :  Here classical queuing theory is extended to represent P2P service systems  Future work directions include 1. Study of M/G/M/M systems with general service time 2. System with different classes of service policy 3. System with different type of job/server correlation 4. System with heterogenous servers 5. More analytical results than just stability e.g average queue length, applying little’s law average service time can be obtained

Queuing Network Models for Multi-Channel P2P Live Streaming Systems

Contents  Infinite server queuing network model  Considers two P2P streaming designs : a) Isolated channel design b) View – upload decoupling  Calculates critical performance measures for both designs.

Introduction  Peers enter and leave video application on longer time scale,peers change channels on slower time scale  Isolated channel : Peers viewing same channel are organized into a swarm, with peers in the same swarm distributing to each other  View-Upload Decoupling : it decouples what it uploads from what it views.

Closed Queuing Network Model  Model for always on or slow churn  Notations : 1. N – set of all peers, n – total no of peers 2. u i – upload rate of peer i 3. J – no of channels 4. r j - streaming rate of channel j 5. v j - server rate of channel j 6. 1/µ j – expected amount of time a peer views channel j 7. p ij – probablity of peer switching from i to j

Let Mj be random variable denoting number of peers viewing channel j System can be viewed as closed Jackson network

Performance Metrics 1. Universal Streaming occurs when every peer is receiving channel it is viewing at the streaming rate of channel 2. Expexted no of Happy Channels Resource Index : б j (M j ) = b j –o j / d j (M j )

Viewing-Uploading Decoupling  Here each stream is divided into substreams  Probability of universal streaming is calculated for each substream from which PS and HC are calculated  Optimal Streaming : For given allocation Nj for channel j,to maximize probability of universal streaming, channel is divided into equal rate substreams

 Asymptotic analysis : Determine under what conditions PS, probability of universal streaming goes to 1, for large number of peers They propose a simple scheme to dimension the group sizes  Similarly for ISO design, expressions of probability of universal streaming and expected number of happy channels are derived and asymptotic analysis done

Peer Churn and Channel Churn  To include peer churn in the model, infinite server open jackson network model is used

Conclusion  Performance of multichannel P2P systems is studied using jackson networks  Model captures essential aspects multichannel video systems such as peers churn,channel switching,b/w heterogeneity  VUD performs better than ISO  Model can be extended to capture heterogeneous peer channel switching patterns