DiffServ Node with Join Minimum Cost Queue Policy: Analysis with Multiclass
’kaffic
D. Manjunath
Ashish Goel
N. Hemachandra
Dept. of Electrical Engg Dept. of Electrical Engg IE & OR Programme
Indian Institute of Technology, Bombay Powai Mumbai 400076 INDIA
{dmanju@,ashgo@ee,nh@}iitb.ac.in
A h b D l f i S e r v is an attractive undldnte for providing relative QoS in the Internet. This Is a h wily amenable to almple
and efi”e
Pricing m-hanlsmk BY Pricing access to a
live QoS, we a n model a MffServ node IU a “Joln Minimum
Cost Queue” in which an n M n g customer (packet or connection) determines the relative cost IU a function of the congestion
in the differentqueues and their access srices and deeldes to take
service fromthat quene for whlch tbe cost Is mlnlmnm. The Paris
Mehn pricing aystem and ita work conserving variant called the
T h p a t i pricing are a n a l y d In this paper In the presence of
mniticiara traffic and for
pricing. fi0of the more interestine observations are that the diautllitv and revenue rate are
nor ~ o ~ o r o nori cconvcx functiou of priie rad (Le revmue rate
is very sensitive lo lhc behavior oflbe delay sensitiveclas.
share of the network resources and the access to each partition
is differentially priced. In [5], [6]the behavior of PhIP under equilibrium conditions is consided and compared with a
uniclass pricing system.
r71 !“poses a work conserving version Of pm called the
liruoati
This takes insDiration from the queue man- svstem.
.
agemat
system at ~ i ~ ~ major
~ t ipilgrimag;
,
centeI in
rndia where it has been operating with remarkable
efficiency for quite some time now. In 171, the social optimality Of Tirupati pricing iS analyzed and it was shown that the
difference between the social cast of the optimaliy priccd system and that of the Timpati system is K e for constants I? and
E. A more practical contribution of [7] was dynamic pricing
using a dynamic programming equation and a reinforcement
I. INTRODUCTION
learning based online pricing algorithm.
In this paper we consider pricing quality through an access
Many pricing models to save the Internet from suffering the
charge
and analyze pricing in a multiclass service system like
“tragedy of commons” have been proposed. Considering the
volume (in bytes, packets or connections) of traffic handled DiffServ where the access charge is a function of the service
by the Internet, it is now clear that these schemes should have class. We abstract an Internet node as a queuing system and
simple implementations and be robust. [I] identifies four cat- analyze the Tirupati and PMP systems with multiclass traffic
egories of Internet charges - access, usage, congestion and and static pricing. Our results are from classical Markovian
quality. It is also generally believed that in this classification,it queuing analysis similar to the analysis ofjoin shortest queue
is computationally expensive to implement pricing shucnues systems from where we borrow some techniques. The paper
for all but the access charges. [2] is an good overview and is organized as follows. In the next section we describe our
models and assumptions. Section I11 describes the analysis
bibliography on Internet pricing issues.
to obtain revenue rate bounds and disutility rate (to
methods
Absolute end-to-end quality assurances is very difficult and
probably infeasible in the Internet. Relative guarantees on a be defined later) approximations for an infinite buffer system.
hop-by-hop basis are more feasible. The Dimerv [3] model Section N presents numerical results from our analysis.
for QoS in the Internet is based on this realisation. In DiRServ
11. JOIN MINIMUM COST QUEUE WITH MULTICLASS
the available bandwidth is divided among multiple classes
CUSTOMERS
statically or dynamically. Network nodes maintain separate
logical queues for each class for each outgoing link and serThe “Join Minimum Cost Queue (JMCQ)” works as folvice them according to the bandwidth sharing policy. Priority lows. A single exponential server of capacity p serves K > 1
queues is the simplest way to provide multiclass service but it queues. The K queues have different join prices p l < pz 5
could lead to starvation of lower priority queues for extended . . 5 pK. We assume Poisson arrivals at rate A. Arrivals
periods of time. A second option would be to have multiple belong to M, M 2 1 classes, and an arrival chooses class
queues and force a certmin arrival profile to each queue to pro- m with r,,, independent of previous arrivals. Let N;(t) be
vide different QoS guarantees to arrivals to that queue. Pric- the number of customers in queue i, 1 5 i 5 K and let
ing seems to be a good vehicle to provide this type of control. N ( t ) = [NI@),
N z ( t ) ,... , N ~ ( t ) l be
* the queue length vecParis Metro Pricing system (PMP) is one such pricing system tor at time t. A class m arrival at time t is informed of N(t-),
for multiclass queues [4]. In the PMP, the network is logically the queue vector ‘just before t’. It calculates its disutility
partitioned in a static manner, each partition is allocated a fixed for queue z according to a function $(m, N ; ( t ) p, i ) and joins
a
.
0-7803-7632-3/02/$17.00 02002 IEEE
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queue i if i minimizes +(m, N,(t-),pj) among all j. Ties in difficult and only approximations and asymptotic results are
disutility are awarded to the lower priced queue. $(.) reflects available for the general problem. See [B] for a survey of the
the sensitivity to price and delay. This policy is clearly indi- analytical models for JSQ. The JMCQ is further complicated
viduaily optimal and [7] showed that for a linear +(.) thc bal- by the existence o f a cost associated with the queues and also
anced JMCQ queue is socially optimal for single class lraffic the presence of multiple classes of customers and we expect
in the sense that the difference between the social cost in this that exact closed form expressions for performance analysis
and the optimally priced system is within
for a constant E . will be difficult. Therefore, we consider an approximate analThe PMP uses a static partition of server capacity among ysis. We will concentrate on two kinds of results - the revenue
K
the K queues - queue i is served at rate p..
p, = p. rate for the server and the disutility rate for the customers.
The evolution of N(t) is a two dimensional birth death proThe Tirupati service system [7] is a work conserving version
of PMP and in its most general form, it uses the genemlized cess. Given N(t), the depamre process is governed by the
processor sharing model to service the K queues where queue service policy - non work conserving PMP or work conservi is allocated a weight w; and receives service at rate p,. Here ing Tirupati. The queue into which an arrival state N(t) will
m.@
U. =
. The JMCO is balanced if the ca- join is determined by its class and its delay sensitivity. Let 6m
a?
L,*,~--PYw~
to be the queue that an arriving class m customer to state [i,31
Dacitv is dmded equally
~.
- .among- all the queues. Thus for all willjoin. ForexampIe,6:, = 1i f $ ( l , i , p l ) S $(l,j,m) and
i, the Tirupati system is balanced if wi = w and the PlvIp is
,:S = 2 otherwise. The transition rate matrix Q = {q;,:ht},
balancedifp; = fi.
We prescribe only a join price and no price for the quantum i,j , k,1 = O , l , . .. is easily determined.
We assume that the Markov chain described by Q above is
of service. This is in the spirit of the Internet in having only
a flat “access” charge, no volume charge and simple to imple- ergodic and a steady state solution E = {T;,} exists. When
ment. The congestion level in terms of the queue length of the system is in state [i,j],a revenue p; is earned if an arrival
each of the queues is posted to enable customers to calculate joins queue i. Therefore, from the Law of Large Numbers, the
their disutility. We assume that the customers obtain instan- steady state rate of revenue generation, ‘R., is given by [9]
taneous congestion information on arrival and that there is no
‘R. =
[~ip11(6:~
= 1) rlpdI(~5;~
= 2)+
delay as might be expected in the context of a an Internet. An
i,j
important conaibution of this paper is that we explicitly in+rzp11(6; = 1) rzpz1(6; = 2)] (1)
volve a model of customer behavior based upon a “disu1:ility”
calculated from the posted prices and congestion levels. We
where I(.) is the indicator function with the usual meaning of
can also consider balking customers and finite buffer queues
taking on a value 1 if the condition is satisfied. Similarly, the
but do not pursue them in this paper.
disutility rate for class m customers, ZJm is given by
Let pim be the join cost of queue i for class m customer.
A convex combination of the queue length (the congixtion
’0, =
[1(C= l)(M (1 - %)PI)+
indicator) and price as a disutility function is simple and efid
fective in capturing price and delay sensitivities for differ+I(6; = 2 ) ( a m j (1 - am)pz)] (2)
ent types of traffic. Thus the disutility function that we: consider is ~ ( mN;(t),p;,)
,
,= atmN;(t) (1 - a;&;,
with We now describe the method to obtain bounds and approxia;,,, E (0,l). The a;, will be called the delay sensitivity of mate results. First consider the revenue rate, R. We borrow
class m. We ftuther assume that the service rates for all c:Iasses the techniques of [IO] to obtain bounds on 77. by considering
are identical and that prices are not class dependent, a rea- a truncated state space. To motivate the truncation, first consonable assumption in the context of the Intemet. Themfore, sider the situation when only one class of customers is present.
a;, = a
, f o r m = l , . . - M andp;, = p; f o r i = l,...K Without loss of generality, let PI < pz. On the N ~ ( t ) - N z ( t )
and $(.) reduces to +(m, N;(t),p i ) = a,N;(t)+ (1-a&;.
plane, an athactor line can be defined such that an arrival to
In the sequel we consider a balanced JMCQ with two cus- the system when the state is on the left of this line will join
tomer classes and two queues, i.e., M = 2 and K = :!. We queue-1 and an arrival to a state on the right of this line will
will also a m m e that p~ = pz = p/2. Without loss of gener- join queue-2. This means that arrivals tend to move the sysality, we assume a l > a ~i.e.,
, class 1 traffic is delay sensitive tem towards the attractor. The attractor line is defined by the
and class 2 traffic is price sensitive. In the next section we equation NZ= N l - ( e ) ( p ~ - p l ) , i.e.,arrivalswillprefer
analyze the system with infinite buffers and no bakiig.
- PI).
the lower priced queue unless N I - NZ > (*)(pz
With multiclass traffic, there will be separate attractors for
111. ANALYSIS OF INFINITE BUFFER JMCQ
each class. Therefore, for states ( i , j )away from the attracThe JMCQ policy is similar to the “Join Shortest Queue” tor we argue that T ; becomes
~
very small and we can con(JSQ) policy that has been extensively studied. The exact per- sider a state space truncated at some distance from the atformance analysis of the JSQ system has been notoriously tractors and suitably modified transition rates. Specifically,
I
.
.-
.
z~;jX
+
~
+
CTijXT,
+
+
+
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-
D,I and F.1 be the corresponding quantities for the JMCQr
Nl
t.
I
..-
.-
.
I
The proof is exactly similar to that in [IO].
Let Sk1be the states on the left threshold line and Skrthe
states on the right threshold line. Let lr; be the stationary
distribution of JMCQu and define 72, as
E!% =
A;{.
[TlplI(6:j = 1) + T 1 n I ( g j = 2)
iiES‘
/
>
e
NI
T-(II.l+llU-Z).(pl-pl)
Fig. 1. Thin lines are arWcrmS for the WO c w t ” class-, 01 > oz.
Thick lincs are mcation thnshalds. Also show are WOcxample stales an
the thnsholds and the Immitiom f”than. Transitiom mslked z BIC not
pnscnt io JMCQ.Tmmitionr marked d l and d l will &e the new system
“leave” S. T-itioN
di and dz are dropped in JMCQ.. Transitions 11
aod Iz with rates d l and dg nspeaivcly are inmduced far J M C Q I .
Along similar lines we can define 721.
Proposition 2: 72, 2 72 2 72,
Proofi We sketch the proof of the first half of the above
inequality. The JMCQ” and JMCQr behave identically to
the JMCQ for states between the two threshold lines. Furwe will consider the state space of the JMCQ truncated to ther, when the state is on a threshold line, the JMCQ, and
contain only those states for which 0 5 IN1 - Nal < T, JMCQl behave identically to JMCQ for arrivals and we
T2
e ) ( p z -p,). Twill be called the truncation need to consider only departures corresponding to transitions
threshold. Let S denote the state space of the original JMCQ dl and d2 of Figure 1.
system and S’the state space obtained after truncation.
Disallowing transition dl increases the disutility of the
We define systems JMCQ, and JMCQI over S‘, like in
lower priced queue leading subsequent arrivals to join the
[IO]. These give us upper and lower bounds on the mean delay
higher priced queue with a higher probability, hence inand the mean number in the system for the JMCQ. First concreasing the revenue on an average. While the system is
sider system JMCQ, and let Q, = {q$} be its transition
on the left threshold line, revenue pl is accumulated into
rate manix which is obtained f “ Q as follows. The hansi72,at rate in Eqn. 3.
tion rates h m the states that are between the two truncation
Similarly, dz is disallowed in J M C Q , and from Eqn. 3,
threshold lines will be the same as that in the JMCQ system.
accumulates revenue p~ at rate f while the system
In the states on the truncation threshold line, an arrival will
is on the right threshold line. This more than compenmove the state space towards its amactor and will not cause
sates for the loss in revenue due to increase in disutility
the system to go out of S’. In a state on the threshold line, a
of queue 2 .
departure is disallowed. This means,
The JMCQl and JMCQ, systems can be solved as follows.
Rename the states in S’as follows: (NI,Nz) is renamed
q$i1 = q,j:k, V ij, kl E S’
a s ( N ~ - N z , N z ) f o r N ~= Na,...Nz+T . Sincethesystem
Now consider system JMCQl and its transition rate ma- is a two-dimensional birth death process, the non zero transitrix Qr = {&,} which is obtained from Q as follows. As tion rates ate only between “adjacent” states and the transition
with JMCQ,, the transition rates from the states between the rate matrix for the JMCQr and JMCQu systems will have
threshold lines are the same as that in the JMCQ system. Once the blockttidiagonal structure of a quasi birth death process
again, as with the JMCQ,, we need to consider only ‘the and the matrix geometric techniques [ 1 I] are directly applicatransition rates corresponding to departures from states on the ble. For more details of this mapping see [IO].
truncation threshold line. These are modified such that a deIV. NUMERICAL
RESULTS
parture will take away a customer from the other queue. This
means
We Dresent some numerical results of ow analvsis of the
i f i j is On left b s h o l d ; k = i - 1, = j - 1 PMP and Tirupati variations of the IMCQ and analyze the deqij:,-l
ifij is on right threshold; k = i - i, 1 = j - pendence of revenue and disutilities on the delay sensitivities
q,j:; j-l
of the customer types and the price vector. We will use pl = 0
q.a:kl
otherwise
and p2 = p. Thus price variation will essentially mean a variaLet D,, the mean delay for class-m traffic and
the tion ofp. Also, unless explicitly mentioned, AI = A2 = 0.5A,
mean number in queue i in the JMCQ, system. Also let the amvals are equally likely to belong to either traffic class.
(e
+
.
r;u
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%
2
1.5
1
0.5
n
0
5
I5
20
PlTce of queue 2 (PI
10
30
Rie of 4yIys 2 @)
Fig. 2. Comparing wmyc m e f"J M C Q , and JMCQi Bgiinst resub f"simulation. ' l l ~ h c mmcatioo threshold is chosen at two levels U)the
nght of the right a n " line, i.e., T = flp1 2. Computation LSCS 100
stage of the QBD prccess. X = 0.8.01= 8.8 and (12 = 0.3.
+
4
Further, the results are primarily for the Tirupati case unless
explicitly mentioned.
First, consider the tightness of the hounds on the nvenue
rate. Figure 2 shows the upper and lower bounds on the R
from analysis and that obtained from simulation. Observe
that the hounds are very tight, especially for higher vilues of
p. Also observe that although the revenue rate generedly increases with increasing p reaches a "maximum" and then decreases, it is neither monotonic nor a convex function of p.
This means that R does not have a unique maxima.
Note that we have not established that the disutility rates
obtained from either the JMCQu or the JMCQi systems are
bounds on the actual disutility. We will use the JMCQ, system to obtain the disutility rates and treat the results as an approximation of the actual value rather than as an upper bound.
Next we compare the revenue rate in the Tirupati and PMP
pricing models. In PMP, if X > pl.queue 1 will never empty
and the system will always operate near the equilibrium lines.
Increasing p will only move the equilibrium lines and 72 will
be an increasing function of p. This is shown in Figure 3 for
X = 0.7. However, when X 5 pl,i.e., X = 0.4, the revenue
rate function of PMP looks similar to that of the Tirupati.
We now consider the effect of price sensitivity on R and
D
' , in the Tirupati queue. Figures 4 and 5 show the revenue
rate as a function of the delay sensitivities of the two classes
of traffic. Although R increases with increasing a,, 1 he following ohservatiotw are interesting.
1) The change in the revenue rate due to change in the a1 is
much more than that caused by a change al. FoI example, even doubling of az from 0.3 to 0.6 does not change
the revenue rate appreciably.
2) The price at which the maximum revenue rate is
achieved is very nearly the same for the differ" values a2 whereas it increases for increasing values of al.
-
3.5-
d
i
'Slio.8'
x
'alio.r- - . x - '*(io,B' .... fl ....
'o,*,,,..*..
3 2.5
-
2 -
x...
I] ..U
0.5
0
0
o"Ufl..
I
0
5
h
.
.
-
-
0.. s . . . ~ ~D..
.
=?=
10
15
20
PlTceotq"eusZ(p)
~ i &
4. R~~~~ me a funaianOfp for different ol;
25
..,
~
30
x = 0.8, O~ = 0.3.
This is because a large fraction of the revenue is derived from
the delay sensitive class and any changes in its behavior have
a more noticeable effect than those in the price sensitive class.
We also observed that the revenue is more sensitive to the delay sensitive class even when (a1 - a2) is small.
Figure 6 shows the disutility rate as a function of p for different a1 and az for Tirupati and PMP systems. Increasing
a1 decreases the disutility rate while increasing az increases
it. Further, it is an increasing function ofp. However, as the
price increases, the disutility rate does not increase beyond a
certain value. This is because all arrivals join queue 1 and
therefore the disutility is less and less a function of p. For
PMP, the disutility increases rather rapidly and linearly.
To study revenue rate as a function of A, we consider two
cases - (i) A is varied with X1 = Az = 0.5X, (ii) X is constant
and X1 and Xz are varied. Figure 7 shows that the revenue rate
behaves much like the mean delay function as a function of A.
This is expected because at high arrival rates, queue lengths
can be significant and the delay sensitive traffic will choose
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:i
18
/
9
'S2io.3'
-
/+
p 2 a . 4 --x-,12;o,5' ..~*..
'#2=0.8' . 0..
0.4
r
0.2
0
10
5
15
20
3.5
i-"a
1HB
2
z
5
10
15
20
25
30
35
40
i
.
c
0
1
.'
,.-•
2.5
0'
30
25
1.5
1
0.5
0
5
10
15
20
25
30
35
40
V. DISCUSSION
We have analysed a DiffServ node with multiclass trafiic
operating as a join minimum cost queue. Specifically,we have
analysed the badeoffs between a service provider's gain (revenue) and the customers' cost (disutility function of price and
delay). Although ow disutility function is simple, the qualitative conclusions are fairly general and not expected to vary significantly for more complex disutility functions. Specifically,
ow observation of the significant sensitivity of the revenue to
the behavior of the delay sensitive traffic.
We are currently investigating the system performance in
the presence of balking traffic and finite remurces at the queue.
We are also considering more than two traffic classes more
sophisticated disutility functions.
ACKNOWLEDGMENTS
We thank Rahul Tandra for discussion on the proofs and for
help in computation of the bounds.
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