Modelling integration of streaming and data traffic - CiteSeerX

Modelling integration of streaming and data traffic F. Delcoigne £ , A. Prouti`ere Þ
½ , G. R´egni´e Þ £

EDF R&D, MRI Department 1, av du G´en´eral de Gaulle 92141 Clamart Cedex, France [email protected]

Þ

France Telecom R&D 38-40, rue du G´en´eral Leclerc 92794 Issy-les-Moulineaux, France [email protected] [email protected] ½ Phone: (+33) 1 45 29 60 18, Fax: (+33) 1 45 29 60 69.

Abstract We study an analytical model to evaluate the performance of the integration of streaming and data traffic in a multiservice network. This model is based on the performance of the M/G/1 Processor Sharing queue with time-varying capacity, for which we provide new and practical results. These results allow the definition of the region where the integration is useful and where the desired QoS is satisfied for both streaming and data traffic. Keywords: Multiservice network, queue with timevarying capacity, stochastic bounds.

1

Introduction

While Internet traffic is currently dominated by TCP controlled elastic data transfers, it is anticipated that streaming applications will rapidly develop and contribute a significant amount of traffic in the near future. It is important therefore to understand the impact this new traffic will have on perceived data transfer performance and to devise appropriate provisioning procedures. In this paper we consider these issues, developing a number of modelling approaches allowing performance prediction for an isolated link integrating both streaming and elastic traffic. We assume the network implements a Diffserv-like architecture, giving head of line priority to packets of streaming flows. This priority, coupled with an admission control to limit the overall volume of streaming traffic, is generally considered sufficient to meet the quality of service requirements of the underlying audio and video applications [5], [7]. We further assume that the remaining bandwidth is fairly shared between concurrent elastic flows. This is a somewhat idealized interpretation of the impact of TCP congestion control that has proved useful in understanding the performance of elastic traffic on a dedicated link [4]. As a measure of performance for elastic flows we consider the expected time to complete the transfer referred to as the expected response time. Our models are based on the fluid flow approximation. Elastic and streaming flows are considered to have a welldefined instantaneous rate. The combined rate of stream-

ing traffic varies exogenously as new flows begin and existing flows end. Elastic flows adjust their rate instantaneously in response to variations in the amount of bandwidth remaining available and to changes in the number of elastic flows in progress. We assume flow arrivals constitute a Poisson process so that the elastic flow response time corresponds to the sojourn time of customers in an M/G/1 processor sharing queue with time varying capacity. A complete analytical evaluation of this stochastic system appears to be impossible. In the present paper we present a number of partial analytical results that illustrate the impact on performance of different streaming and elastic traffic characteristics. An important consideration is that of stability. We assume that overall demand is less than capacity so that the system is ergodic. However, it can occur that the number of streaming flows in progress is such that the current arrival rate of elastic flows is greater than the completion rate. We refer to such a situation as local instability. Since the duration of a streaming flow is typically much greater than that of an elastic flow, it can occur that local instability conditions last for a significant amount of time. In this case, the number of elastic flows and consequently their response time increase significantly. Our models quantify this phenomenon in a number of particular cases, illustrating the impact on performance of the distribution of streaming flow duration. To avoid local instability it is possible to perform admission control on streaming flows to ensure that the service rate for elastic flows is always greater than their arrival rate. This condition is termed uniform stability. We provide upper and lower bounds on the expected response times in this case. These bounds have the significant property of being insensitive with respect to the distributions of streaming flow duration and elastic flow size and prove extremely tight under realistic network conditions. Previous authors have studied the considered integrated services system. Altmann et al. [1] and N´un˜ ez Queija [18] develop Markovian models which can account for flow sizes having a phase-type distribution. These models can provide numerical results (when the number of phases is small) but give little qualitative insight into the impact on performance of particular traffic characteristics. The difficulty arising when the Markovian assumptions are re-

laxed is well illustrated in papers by Boxma and Kurkova [8, 9] and by Nain and N´un˜ ez Queija [17]. These authors study the performance of a queue with time varying service rate in the special case where the latter has just two states with generally distributed duration. Results are derived for the tail behaviour of the workload distribution and the generating function of this distribution, respectively. Generalization to more than two service rates is not obvious. In the present paper, we focus on the derivation of insight providing closed form limit results and approximations. In the next section we describe the fluid model of streaming and elastic traffic integration. We then discuss in Section 3 the phenomenon of local instability and study the impact of streaming statistical characteristics on elastic performance. Section 4 is devoted to a performance analysis in the case of uniform stability; we derive tight upper and lower bounds for the mean response time of elastic flows. In Section 5 we present simulation results to illustrate and validate the analytical results obtained in the previous sections.

2

A fluid flow statistical bandwidth sharing model

In this section, we present a fluid model of a link performing statistical bandwidth sharing between elastic and streaming flows. We first recall results applying to a link dedicated to elastic traffic only.

2.1

Bandwidth sharing with elastic flows only

Consider an isolated link of capacity

dedicated to elastic traffic. Assume flows arrive as a Poisson process at rate 
with a size drawn independently from a common distribution with mean

. Flows are modeled as a fluid whose rate adjusts immediately in response to changes in the number of flows in progress, and we assume bandwidth is shared equitably among flows. The link load is 
 


 

. The number of flows in progress 
behaves like the number of customers in an M/G/1 processor sharing queue [15]. We deduce the following results when 
  :

 
   




(1)



 



 



(2)

2.2

(3)

This fluid model for elastic traffic was introduced by Roberts and Massouli´e [19]. It constitutes an idealized

Accounting for streaming traffic

We now formulate a model to account for the impact of random fluctuations in the number of streaming connections. It is assumed that packets of streaming flows have priority, so that the evolution of the number of streaming flows in progress   , is unaffected by elastic traffic. We assume that streaming flows have constant bit rate so that the value of   determines the amount of bandwidth currently available to elastic traffic. Define the following notation:

  is the stationary probability that   is in state , 

 and 
 




 


are the capacity available to elastic traffic and the elastic load respectively when process  is in state ,







 denotes the mean capacity 






available to elastic traffic,

 





 


stands for the mean offered load of

elastic traffic,

2.2.1 Streaming traffic characteristics The performance of elastic flows depends on the characteristics of the streaming flows defined principally by their rate, their holding-time and the proportion of the overall traffic they represent.

 

More precisely, the conditional mean response time is




  

 



model of bandwidth sharing produced by TCP but accurately predicts the most significant performance characteristics [4, 13]. A key observation is that the above results are insensitive to the flow size distribution. In other words, it is not necessary to know this distribution in order to predict first order bandwidth sharing performance. Note however that the distribution of response times is not insensitive. In [4], it was proven that the above results (1), (2) and (3) still hold for more general flow arrival processes (not only for Poisson processes). This is usefull to take account of the fact that flows are generated in sessions and do not constitute a Poisson process in general.

The rate of streaming traffic depends on the underlying application: less than 64kbit/s for telephony, sometimes more than 2Mbit/s for video. A key factor is the ratio of streaming flow rate to link capacity. The average holding times for most streaming flows is of the order of several minutes. Note that this constitutes a much larger time scale of traffic variations than for elastic flows which typically last a few seconds. We expect the distribution of the holding times for most applications to have a high variance like that of telephony [6].



The future proportion of streaming traffic depends on the considered networking environment. It will probably be greater in mobile networks than in the terrestrial Internet where elastic traffic dominates.

2.2.2 Impact on elastic traffic We again assume that elastic flows share bandwidth equitably with instantaneous adjustment as the number of streaming or elastic flows in progress changes. Assuming Poisson flow arrivals, the number of ongoing elastic flows behaves like the number of customers in an M/G/1 processor sharing queue with time-varying capacity. We assume 
  so that the system is stable. The M/G/1 processor sharing queue with time-varying capacity is not insensitive, i.e., performance strictly depends both on the flow size distribution and on the parameters describing the streaming traffic process. There are no simple exact results analogous to those of the classical processor sharing queue recalled above. In the following sections we derive practical and accurate approximations valid in two distinct regimes: Uniform stability Whatever the state  of   , available capacity for elastic traffic is sufficient to meet demand:


   

Local instability For some states  of   , available capacity for elastic traffic is not sufficient to satisfy demand:  such that 
  

3

Local instability

We now derive a number of analytical results providing insight into the impact on elastic flow response times of streaming traffic characteristics when the system is locally unstable. We assume streaming traffic can be described as a semi-Markov process, i.e., when   enters state , it remains there during an arbitrarily distributed time  . Let  denote the distribution of   :           At the end of this period of time,  jumps to state  with probability   where    is the transition matrix of a stable Markov chain with finite state space. For the sake of tractability, the size of elastic flows is assumed to be exponentially distributed with mean

  
. Note that even with the latter assumption, the problem of evaluating the mean response time remains open.

3.1

A rough lower bound for elastic flow response time

The following proposition shows that the mean response time increases at least as fast as a linear function of the mean and the variance to mean ratio of the length of an

instability period. This is particularly significant when the distribution has a heavy tail or when stream traffic evolves at a time scale much greater than that of elastic traffic. Proposition 1 The mean response time of elastic flows satisfies the following inequality:

 


 

 

   

  
 




(4)

Proof: if  denotes the time elapsed since the last change of   , we have

            
   
  Moreover, if    represents the number of flows at time  in an initially empty M/M/1 queue with capacity

 , then for all ,  

      
    




  Hence,


 1I



 

  

   
    



which implies


 1I



 










 
   




 







 







 

Summing over instable states and applying Little’s formula give the bound of the proposition. Remark: using refined results on the transient behaviour of the M/G/1 PS queue and in particular formula (2.6) of [14], the following bound can be derived for generally distributed elastic flow sizes:

 


 

   

  
  

(5)

where  is solution of the fixed point equation:

 

  









  





 

Proposition 1 gives a very general bound for the mean response time of elastic flows in the situation of local instability. In the following we show that it is possible to derive more precise evaluations in some particular cases.

3.2

Model with service interruptions

We first recall a result derived by N´un˜ ez Queija [18] giving the expected response time in a system where elastic traffic is subjet to random service interruptions. In this case, two states are allowed for process   : in state “on” (  ), the available capacity for elastic traffic is 1, and in state “off” (   ) no elastic traffic is served. When the on-period   is exponentially distributed, the stationary distribution of 
is known and we have:

Proposition 2 The conditional mean response time is given by:

 


 




     


 



     

  



 


  
  

   






Note that the first term in this formula may be interpreted as the mean transfer time for the processor sharing queue with capacity  which is equal to be the mean capacity

in this case. This formula highlights the precise dependency of the mean response time on the first and the second moments of the instability period: it increases linearly with
 
  
  Var   
 .

 

3.3

This result suggests that the mean response time is roughly proportional to the mean duration of the instability period (i.e. proportional to ).

3.4

It proves difficult to extend the analysis of the quasistationary regime to a system with more than two streaming states. In the following we give approximations valid in heavy traffic, i.e. when 
tends to 1. The proof of the following result is not presented here for the sake of concision but may be found in an extended version of the paper [12]. Proposition 4 When 
tends to 1,



 











Quasi-stationarity

As mentioned in subsection 2.2.1, elastic and streaming traffics evolve at different time scales: the time required to transfer a web page should be far shorter than the mean duration of an audio or video application. The number of ongoing elastic flows 
evolves rapidly with respect to process  describing streaming traffic. It is intuitively obvious that this difference in time scales is a supplementary factor of degradation of elastic performance, since the local instability periods are relatively very long. To capture this effect, we introduce a renormalized sys  tem  
, where the speed of streaming traffic evolution is multiplied by parameter :    remains in state  during a period of time   . Note that in equilibrium,  and  are identically distributed so the load remains unchanged. We analyze the behaviour of  
in the quasi-stationary regime when  tends to 0. A detailed study of such a system was made by Choudhury et al. [11]. It was shown that process   
     converges in distribution, when  tends to 0, to a certain limit process. This analysis only gives a convergence result. To de termine the quantitative behavior of 
when  tends to 0, we have directly computed the generating function  of 
for the particular case where two states for   are allowed. Assume that in state 1 the system is stable (
  ) while in state 2 the system is unstable (
  ) and that periods of stability are exponentially distributed. Tedious computations lead to the following proposition.





Proposition 3 When  tends to 0,



     


 where

   





 




   




 






where

(7)

is given by

 Var   



     
   !
 ! (8)









and where the numbers linear system: for all ,

!





















 



! are solutions of the following





!  







 












(9)

Proposition 4 again states that the expected response time increases in proportion to the mean and the variance to mean ratio of the duration of the instability periods.

4

Upper and lower bounds for elastic traffic performance in uniform stability

In this section, we consider the uniform stability regime and derive lower and upper bounds for the performance of elastic traffic. These bounds are insensitive, i.e. the mean response time of elastic flows does not depend on the elastic flow size distribution or on the detailed characteristics of streaming traffic.

4.1 (6)

Heavy traffic analysis

Comparing flow response times

The lower bound is derived assuming that elastic traffic is served at a constant rate, equal to the mean capacity left by streaming traffic

. In other words,   is assumed to evolve so fast that elastic traffic is not affected by its changes, parameter  defined in section 3.3 tends to . We refer to this case as the fluid regime.  
and


 denote the number of simultaneous elastic flows and

5

the response time in stationary regime, respectively. They are characterized by (1), (2) and (3) with

replaced by

. The upper bound is derived under the quasi-stationary assumption: 
evolves rapidly with respect to   and when the latter is in state , 
attains its stationary regime described by the measure of an M/G/1 processor sharing queue of capacity

 . Parameter  tends to 0.   In this case, if 
and
denote the number of simultaneous elastic flows and the response time respectively, we have:

We consider a link of capacity
  Mbit/s shared by both types of traffic. Streaming calls are supposed to have a constant rate "   Kbit/s. We assume that a capacity

 

Kbit/s is reserved for data traffic and that streaming traffic load    
 " 
is fixed at 0.25 (so that the blocking probability for streaming calls is 2%). The mean elastic flow size is

   Kbytes. With these assumptions, the system is in uniform stability for 
 . Remark that the fluid and quasi-stationary bounds derived above are very close and tight in uniform stability. These bounds are all the more tight that the number of possible simultaneous streaming calls is large (in our example, this number is small, equal to 10).

   
    
  
 


   








































 

 






(10) (11) (12)

Our objective is to compare the general process to the fluid and quasi-stationary processes. To this end we use the notion of stochastic ordering, see for example [2]. Definition 1 "#-ordering. Let $ and  be two random variables on Ê , $   if and only if for all convex and increasing functions % Ê  Ê,
% $  
%  . It follows, in particular, that $    implies all moments of $ are less than the respective moments of . The following two theorems are proved in Appendix A. They are based on the notion of superadditive ordering introduced by Rolski [20] to prove Ross conjecture in some cases. Theorem 1 Let &
denote the stationary workload of the queue. The following inequalities hold:

&
  &
 &
 

(13)

Theorem 1 is true in general for the G/G/1 queue with variable service rate for any work-conserving service discipline. It thus provides bounds on the workload of the elastic queue when the service rate varies due to streaming traffic. It proves much more difficult to derive similar bounds for the expected response time. The following theorem applying to the case of an exponential size distribution is proved in Appendix A using the equivalence of Processor Sharing and FIFO disciplines. Theorem 2 When elastic flow sizes are exponentially distributed, we have:


 
 
 



(14)

Applying Little’s formula we deduce:







 



(15)

In the next section, we illustrate the analytical results we derived so far giving an example where a fixed admission control is applied to streaming calls.

Simulation results

5.1

Impact of the mean and the variance of streaming calls duration

Figure 1 plots normalized throughput '

as a function of load 
. Elastic and streaming flow sizes are exponentially distributed. This figure demonstrates that performance in the uniform stability region is good and does not depend on the mean streaming flow duration. On the contrary, when elastic load 
exceeds 0.68, local instabilities occur and elastic throughput falls away. The decrease is more abrupt as the mean or the variance of flow duration increases reflecting a more pronounced impact of local instability. To increase the flow duration variance, we use several hyperexponential distributions defined by



 

 

#     #  ! !  with associated variance Var    ! !

   . Increasing the parameter ! increases the variance of  .

We also can quantify the degradation by computing, for different mean flow durations, the asymptotic slope of '
given in formula (7) under heavy traffic conditions: '
 

.


  

30 sec 0.22

1 min 0.13

3 min 0.05

10 min 0.016

1h 0.003

The values in the table should be compared with 0.75, the corresponding figure obtained for a simple queue in heavy traffic with fixed capacity

.

5.2

Impact of the nature of the elastic flow size distribution

Finally, in Figure 2 we compare the realized throughput when the variance of elastic flow sizes increases. The system is again insensitive in case of uniform stability. In local instability, performance is better when elastic flow sizes are more variable. This is a significant observation

0.8

0.8 Fluid regime QS regime No integration 1 min 3 min 10 min 1h

0.6 0.5

Fluid regime QS regime No integration Deterministic Exponential Hyper exponential a =5 Hyper exponential a = 50

0.7 Normalized flow throughput

Normalized flow throughput

0.7

0.4 0.3 0.2 0.1

0.6 0.5 0.4 0.3 0.2 0.1

0

0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Elastic load

1

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Elastic load

1

0.8 Fluid regime QS regime No integration Exponential Hyper exponential a = 5 Hyper exponential a = 20 Hyper exponential a = 50

Normalized flow throughput

0.7 0.6 0.5

Figure 2: Impact of the elastic flow size distribution Streaming lows exp with mean
    min

0.4 0.3 0.2 0.1 0 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Elastic load

1

Figure 1: Impact of the average (above) and the variance (below) of streaming flow duration.

since it is well-known that the distribution of elastic flow sizes is indeed extremely variable. The above results all confirm that elastic flow performance is largely insensitive to the characteristics of streaming and elastic traffic as long as the system remains in the regime of uniform stability. This is an extremely useful property in that it suggests that provisioning does not depend on the precise characteristics of applications which can change quite radically over time.

6

Conclusion

We have proposed a fluid model to evaluate the performance of a network link integrating streaming and elastic traffic. Assuming Poisson flow arrivals 1 , priority service for streaming traffic and fair bandwidth sharing for elastic flows, the underlying stochastic system is an M/G/1 Processor Sharing queue with time-varying capacity. The main contribution of our work is to give new and practical analytical results for this system. The analysis highlights the impact of local instability 1 In

practice, the flow arrival process is not Poisson, because flows are generated in sessions [4]. In future works, the analysis should be generalized to more general arrival processes.

periods during which elastic flow performance can deteriorate rapidly since the arrival rate of new flows exceeds the flow completion rate provided by currently available link bandwidth. Our models show that the expected response time increases linearly with both the mean and the variance to mean ratio of the duration of the local instability periods. The effect of such periods may thus be significant when streaming flows last much longer than the typical elastic flow or when the distribution of the streaming flow duration is heavy tailed. Performance is generally much better and more predictable if the system is uniformly stable, having no periods of local instability. In this case we have derived a good approximation for the expected response time using a quasi-stationary approximation. In fact, the latter has been shown to constitute an upper bound. A lower bound is also derived by analysing an alternative fluid regime. Comparison of the two reveals that they are generally tight. The insensitivity of the bounds implies that the performance of the integrated system is also largely insensitive to the detailed characteristics of both streaming and elastic flows. As a practical consequence, we propose the following method for capacity provisioning when the known characteristics of demand are the volume of elastic traffic, the volume of stream traffic and the bit rate of a stream flow. The amount of bandwidth necessary to meet the blocking probability requirement of streaming traffic determines the admission control threshold. The supplementary bandwidth

necessary for elastic traffic must then be determined to satisfy two conditions: it must ensure uniform stability (i.e. be greater than elastic demand 
( 
) and it must be sufficient to meet the required condition on the expected throughput of elastic flows as estimated using the quasi-stationary bound. Note that it would be impossible to control expected throughput if the system could exhibit local instability since performance would then depend on detailed and generally unknown traffic characteristics, as shown by the results in Section

are nondecreasing in  and tend to &
and &
respectively. Now for all increasing and convex function % , by the Lebesgue monotone convergence theorem, 

3. An alternative and less conservative provisioning procedure could be applied if the network were able to perform flow level admission control for both streaming and elastic flows, as envisaged in [3].




, *

*       
% &  
%  , * 
 
  , *

    

Acknowledgment. We are grateful to James Roberts for his very valuable comments and advice on the work presented in this paper.







 

Stochastic bounds - Proofs of theorems 1 and 2



We now

















*





  , *
   


 











 














random Jensen’s


















   / *
 *   



 

 




 
%  , * 

.
  , *

.  
   %   , * 


  , *

   


  / * 



and

  , * 






Taking the expectation w.r.t. the arrival process gives the desired inequality. Proof of Theorem 2: again, the first inequality is given in [2], chapter 4. The proof of the second inequality is based on the following representations of the stationary number of flows in progress:

The sequences

 





 

is superadditive.

















 

  -#  %  % #    % #    % #





 

Lemma 2 let % Ê Ê be an increasing convex function and % Ê Ê,       + a sequence of affine functions. Assume that each function %  is decreasing w.r.t. each coordinate. Then

 









%  , *


* 
     , *


*



  , *   , * 

*    



  

inequality:

For all superadditive



%   , * 

* 
  , *

*         
  write
*  




. 

, where . is a uniform variable on     independent of

. By

 


 . Ap, *








 
%
*    
*     
%
*    
*  





 















%  , *


* 
     , *


*

Proof of Theorem 1: the first inequality is given in [2], chapter 4. Let us turn to the second inequality. Let

 denote the stationary process representing the actual available capacity for elastic traffic at time . Let

)  

 , and

*   


  where    . Write +    . Let , * be the amount of elastic traffic arriving to the server between times   and  . We need the following lemmas found for example in [21] and [10] respectively.








Let

denote the expetation w.r.t. 

  plying Lemma 1 and Lemma 2 with %  #  # , we have

Definition 2 superadditive function. % Ê  Ê is superadditive if and only if for all #   Ê  , % #   % #    % # %  , where #     #   and #     #   .

Lemma 1 (Lorenz inequality). function % ,



 

 

Theorems 1 and 2 are both based on superadditive ordering and the Lorenz inequality. The method is similar to that used by Rolski [20].











A




% &
  
%  , *


* 









  / * 




 *


   / *


  *   

 








  *

where / is a Poisson process of intensity 
,  is a doubly stochastic Poisson process of intensity measure

 

, and for a fixed

 ,  is a Poisson process of intensity

 
. Then for all increasing and convex function % ,



[4] S. Benfredj, T. Bonald, A. Prouti`ere, G. R´egni´e, J. Roberts, Statistical bandwidth sharing: a study of congestion at flow level, in: Proceedings of Sigcomm, 2001.




% 
  
%  / *
 * 


/ *
 *      
%   / * 

  *      
 / *

  *  




% 










   



 

Let






0 #   %  #
   







#



 


 for

#   Ê , and


   
 
0/ # . - #  
    

Lemma 3

- is superadditive. See [20] for example.

It follows from the Lorenz inequality:






%  / *
 * 
     / *
 *  






- 

*   

* 
- 

*   

* 






%  / *
 * 
     / *
 *   

The proof is completed using the same argument as in the proof of Theorem 1.

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[18] R. N´un˜ ez Queija, Processor-Sharing Models for Integrated-Services Networks, PhD Thesis, Eindhoven University of Technology, 1999. [19] J. Roberts, L. Massouli´e, Bandwith sharing and admission control for elastic traffic, Telecommunication systems, 2000. [20] T. Rolski, Upper Bounds for a Single Server queue with Doubly Stochastic Poisson Arrivals, Mathematics of Operation Research, 11 pp 442-450, 1986. [21] A. H. Tchen, Inequalities for Distributions With Given Marginals, Annals of Probability, 8 pp 812827, 1980.