1

Spot and Derivative Markets in Admission Control

Nemo Semret� and Aurel A. LazarDept. of Electrical Engineering, Columbia University,New York, NY, 10027-6699, USA.fnemo, aurelg@comet.columbia.eduhttp://comet.columbia.edu/publications

We propose a new approach to pricing of capacity in service systems with blocking,using spot and derivative market mechanisms. A second-price auction among arrivalsgrouped in batches gives rise to the spot market of usage charges.A reservation guaranteeing access for an arbitrary duration with a usage price below

the bid can be made at any time before or during service, thus eliminating the risk {inherent to the spot market { of being dropped before service completion. We de�nethe reservation as a hold option, which is analogous to derivative �nancial instruments(e.g. options, futures) integrated over time. Based on a heavy-tra�c di�usion model forthe corresponding two-stage queueing system, we compute the reservation fee as the fairmarket price of a hold option.We validate this approach with simulations driven by a real tra�c trace at a dial-up

Internet access modem-pool.Keywords: network admission control, market pricing, reservation, options, di�usion

models, stochastic di�erential equations

1. INTRODUCTION

In modern networks, due to the great variability of demand, there is considerable inter-est in pricing network resources using market mechanisms. In earlier work, we proposed apure market approach for bandwidth pricing, where allocated capacities may vary duringthe lifetime of the ow, as players compete for resources through an auction game [6,7].In this paper, we propose a mechanism for circuit switched calls, wherein calls are

admitted or rejected at (or soon after) their arrival time, and if admitted, get a �xed allo-cation of capacity, and have the option of securing the resource at a guaranteed maximumprice for a guaranteed minimum duration. Thus the charge has two components:

� a market-based usage charge, where the user continuously pays the \instantaneous"market price (determined by second price auctions among recent arrivals); if themarket price exceeds a user's bid price, that user is dropped unless

� upon arrival, the user pays a reservation fee (buys an option contract), which giveshim the right to buy the capacity at any time in the future up to a speci�ed duration

�Corresponding author.

2

at his bid price. The user pays the market price as long as it is below his bid price. Ifat some point during the call, the market price exceeds the bid price, then the userautomatically exercises the option, i.e. remains connected while paying no morethan the bid price.

In other words, we introduce a \derivative" instrument to reduce the uncertainty inher-ent in the \spot" market mechanism. Naturally, this contract (or reservation) must itselfbe sold for a \fair" price. In the context of �nancial markets, the fair price is calculatedwith the Black-Scholes approach, which is based on the idea that the option must bepriced such that a perfectly hedged (i.e. riskless) combination of the derivative and theunderlying equity will provide (locally in time) the same expected return as a risk freesecurity[3] (otherwise the derivative would present an arbitrage opportunity, i.e. unfairadvantage, which would be exploited until its price rises).In our context, the contract di�ers in that, rather than the right to buy once at a given

strike price, it gives the right to buy repeatedly over a given duration. The \fairness" weseek is that the reservation be priced in a way that re ects the probability that the systemwill become busier during the lifetime of the reservation, in order to avoid individuallyrational but socially sub-optimal behaviour. Speci�cally, if the reservation is too cheap,the sel�sh users connecting at low periods will make reservations with an excessively lowmaximum usage price for a very long time, to avoid rejoining when the prices are higher.A simpli�ed form of this arbitrage is commonly observed with at-rate priced dial-upInternet access, where some users remain logged on for very long periods of time even ifthey are not using the network, in order to avoid the risk of later getting busy signals.A further di�erence from usual options is that, rather than the standard geometric

Brownian motion model of e.g. a stock price, we have an underlying spot market processthat is derived directly from our queueing system, via a heavy-tra�c approximation whichleads to a di�usion process.From the point of view of the user, this pricing mechanism is simply an initial reservation

fee plus a per-minute (for example) usage charge. The novelty in the networking contextis that both are market prices.In Section 2, we derive a heavy-tra�c di�usion approximation model for the queueing

system. Using that, in Section 3, we derive the corresponding model for the spot marketprice. Then in Section 4, we introduce our formulation of the reservation as a derivative�nancial instrument. Finally, in Section 5 we present some simulation results using areal trace of tra�c at a dial-up Internet access modem pool. Due to space constraints,detailed calculations are omitted, and can be found in Chapter 4 of [7]. Although notstrictly necessary, a layman's grasp of �nancial markets is helpful in reading this paper.

2. QUEUEING MODEL

2.1. Preliminaries

Customers arrive in a Poisson stream of rate �, with i.i.d. bid prices distributed ac-cording to a distribution F (density f). Call durations are exponentially distributed withmean 1=r. The queueing system is shown in Figure 1. It consists of two stages, withbu�ers of size B and C respectively.The �rst stage consists of a second price auction, where the winners enter the second

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batch ouputevery τ

B lines

λ

arrivals Poisson ~ λ

n occupied

~exp/µ

πC lines

βB

of size m ~1<= m <= B

βmbid price ~ F

holding times ~ exp/r

m-(C-n) lowest bidders

C-n highest bidders

Figure 1. Queueing Model

stage, and the losers leave the system. Speci�cally, the �rst stage has an exponentiallydistributed service time � with mean 1=�. At a service completion instant, let m be thenumber of customers in the �rst stage at that time and n the number in the second stage.The m �rst stage customers are ranked according to their pid prices. The (C�n) highestbids are accepted into the second stage, and the remaining m � (C � n) are dropped.The bid price of the highest dropped bid de�nes the new spot market price, which is validuntil the next batch. If no customers are dropped, i.e. m � C � n, then the spot marketprice is zero.The second stage is a C server queue with no waiting room, where each customer has

an exponential service time (call duration) of mean 1=r. The arrivals are according toa batch Poisson process, with rate �, and batch sizes distributed according to f�mgBm=0.Let � = �=�. For 0 � m < B,

�m4= Pfm arrivals in �g = �m

(1 + �)m+1; (1)

and

�B =1X

m=B

�m�

(� + �)m+1=

�

1 + �

!B: (2)

The steady-state distribution of the occupancy of the second queue can be shown to beof product form, given explicitly in [7]. However, for the purpose of pricing the reserva-tions, we need a (probabilistic) model of its transient behaviour in the future, given thecurrent price.

2.2. Di�usion approximation

We develop a tractable approximation for the queue occupancy process Nt, by scalingthe system following the well-known heavy tra�c approach [4,1,8,9,2], applied here to a

4

system with batch arrivals.

Let �4= � �m

r, �m

4= IEM =

PBm=0m�m, and v2

4= IE(M2) =

PBm=0m

2�m. Consider a

sequence of scaled systems, indexed by l = 1; 2; : : :, with capacity C(l) = l� + pl�,

where is arbitrary; and batch arrival rate �(l) = l�. Let

Z(l)t

4=N

(l)t � l�pl�

; (3)

the translated and scaled version of the queue occupancy of the l-th system. The driftand di�usion coe�cient respectively are

a(l)(z)4= lim

�!0IE

24Z(l)t+� � Z(l)t

�jZ(l)t = z

35 =

( �rz if z <

�r( +pl�) if z � ;(4)

and

�(l)2(z)4= lim

�!0IE

264�Z(l)t+� � Z

(l)t

�2�

jZ(l)t = z

375 =

(r(1 + v2= �m+ z=

pl�) if z <

r(1 + =pl�) if z � :

(5)

The basic idea is that as l ! 1, Z(l)t ) Zt, (converges in distribution). Zt is the

di�usion process which solves the stochastic di�erential equation

dZt = a(Zt) dt + �(Zt) dWt; (6)

where Wt is a Brownian motion process.Note that, for <1, the o�ered load �(l)=(C(l)r) = (1 + =

pl�)�1 % 1, which makes

it a \heavy tra�c" approximation.The coe�cients a(:) and �2(:) are obtained by letting l!1 in (4) and (5) respectively.

In the original approach of [4], Z(l)t ) Zt holds if a(:) and �

2(:) are continuous, which isclearly not the case here, since both have jumps at z = . However, based on a theoremof Borovkov [1], Whitt [9] proposes a conditioning heuristic to derive approximate steady-state blocking probabilities for the G=GI=s=0 (i.e. s servers, no waiting room) queue fromthe di�usion approximation for the corresponding G=GI=1 system. Whitt goes furtherto say that, for exponential service times, the heuristic is also applicable to the di�usionprocess itself. Our approach here will be to apply the heuristic to the di�usion processitself, conditioning on Zt � .

3. DIFFUSION MODELS OF THE SPOT MARKET

3.1. Spot Price

Suppose that at time t, the queue occupancy is Nt = n, and a batch of size m arrives.The admission decision is made, and a new spot price results. In this section, we want toderive a model which will tell us in some sense the future evolution of this price.Consider the mechanism by which the price arises. Of the m new customers, those with

the C�n highest bid prices will be admitted, and the spot price will be the (C�n+1)-thhighest bid price. Recall that the bid prices are i.i.d., with distribution F , which weassume to be smooth. The probability that the price is x is the probability that in m

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draws, one will equal x, and of the remaining m� 1, C � n will be greater than x, and(m� 1) � (C � n) will be less than or equal to x, which is

fn;m(x)4= mf(x)

(m� 1)!

(C � n)![(m� 1)� (C � n)]![F (x)]m�C+n�1 [1� F (x)]C�n

=m!

(C � n)!(m� 1 � C + n)![F (x)]m�C+n�1 [1 � F (x)]C�n f(x):

The distribution is

Fn;m(x)4=Z x

0fn;m(y) dy =

Z F (x)

0

m!

(C � n)!(m� 1� C + n)!um�C+n�1(1� u)C�n du; (7)

where the last equality comes from substituting the previous expression, and making thechange of variables u = F (y); du = f(y) dy.Let us consider a \�rst-order" approximation, the expected spot price given the occu-

pancy n and the size of the batch arrival m:

n;m4=Z 10

[1� Fn;m(x)] dx =Z 10

Z 1

F (x)g(u; n;m) du dx =

Z 1

0F�1(u)g(u; n;m) du; (8)

where

g(u; n;m)4=

m!

(C � n)!(m� 1 �C + n)!um�C+n�1(1 � u)C�n: (9)

3.2. Di�usion model of the spot price

For the purpose of pricing a reservation beginning at time t, we would like to characterizethe future evolution of the market price, in terms of what is known at or just before t,namely the occupancy Nt. Thus, it is natural to take the process

(Nt) = IE [ Nt;m] =BX

m=0

�m Nt ;m: (10)

Remark: We prefer the above expression to one that uses the information in the batchthat arrives at time t, since computing the option prices based on the new arrivals wouldadd to the delay in the admission decision. By not waiting for the new information, wecan use the time in between batches for computation of option prices. Of course the spotprice results from the auction among the new bids, but that is a very simple computation.Now, we approximate the price by a sequence of functions of the centered and scaled

process Z(l)t ,

P (l)(z)4= (z

p�l + �l): (11)

Then, Ito's rule (see e.g. [5]) along with (6) yields the stochastic di�erential equation forP :

dP(l)t =

"dP (l)

dza(l) +

1

2

d2P (l)

dz2�(l)2

#dt+

dP (l)

dz�(l)dWt: (12)

6

Since

P (l)(z) =BX

m=0

�m

Z 1

0F�1(u)g(l)(u;

p�lz + �l;m) du; (13)

where g(l) is de�ned as g with C(l) instead of C, it follows that

dP (l)

dz(z) =

BXm=0

�m

Z 1

0F�1(u)

@

@zg(l)(u;

p�lz + �l;m) du; (14)

and

d2P (l)

dz2(z) =

BXm=0

�m

Z 1

0F�1(u)

@2

@z2g(l)(u;

p�lz + �l;m) du: (15)

Now (12), along with (4), (5), (14) and (15), constitutes a di�usion model for the spotmarket price Pt, as l!1.

Proposition 1 We can approximate (12) by dPt = 0 when z < , and

dPt =BX

m=4V 2

�m

�rC

�� 1

mPt +

m� 1

mK

(m)1 +

1

2K

(m)2

�dt +

prC

�1

mPt �K(m)

1

�dWt

�; (16)

when z = . For each m,

K(m)1 =

Z w(m)1

w(m)0

F�1(m

m+ w2)w2 +m

m

s2

�e�w

2=2 dw

K(m)2 =

Z w(m)1

w(m)0

F�1(m

m+ w2)

w2 +m

m

!2 �1� 1

w2

�s2

�e�w

2=2 dw:

V is a constant related to the accuracy to which we want to evaluate g, and can typicallycan be taken to be 1.

Proof: See [7], Chapter 4. 2

Remark: z = corresponds to Nt = C, i.e. the system is full. Thus, we have atwo-regime model, where the spot price remains constant (= 0) when the system is belowcapacity, and varies according to (16) when the system is full. This corresponds to whatone would intuitively expect from the auction mechanism.

4. COMPUTING RESERVATION FEES: THE DERIVATIVE MARKET

Taking the current market price Pt as initial condition, the solution of (16) provides astochastic model of the future prices fP�g��t. This solution is unique in distribution [5].From that, we now determine the price of a reservation, arriving at time t, with a holdingtime T , and a bid price p. The value, or \fair price", at time t of an option to buya security for a \strike price" p at a speci�c future date � � t is, taking the \risk-freeinterest rate" to be 0,

��4= IE(P� � p)+: (17)

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That is called a European option [5,3]), as opposed to, e.g., an American option, whichis the right to buy at any time between t and � . In our context, this concept must beextended in the following straightforward manner: we de�ne the reservation as a holdoption, a new kind of derivative instrument which is an option to buy repeatedly at everytime instant from t to t+ T . Thus the reservation fee should be

�(t; T; p)4=Z t+T

t�� d�: (18)

If K1 = K2 = 0, �� is given explicitly by the Black-Scholes formula [3]. For the moregeneral form (16), which we re-write as

dP� = (AP� +K3) dt+ (DP� +K4) dWt; (19)

for � � t, with known initial condition Pt, the solution is (see [5], Section 5.6)

P� = S�

�Pt +

Z �

t

K3 �DK4

Sudu+

Z �

t

K4

SudWu

�; (20)

where

S�4= exp

��A�D2=2

�(� � t) +

Z �

tD dWu

�: (21)

Knowing the set of distributions Pt4= fIP(P� � xjPt);8x : � > tg, we can evaluate �� ,

for � > t and the reservation fee � follows.

5. SIMULATIONS

We present a brief overview of simulations of the pricing mechanisms presented in thepreceding sections, applied to dial-up Internet access (a fuller treatement is given in [7]).Our data set consists of all the connections that were made to the Columbia Universitymodem pool between April 4th and May 10th 1998. There was a total 669,994 calls, from9451 di�erent users, with an average inter-arrival time of 5.59 seconds, and an average callduration of 1299.3 seconds. The real system had 312 lines, but to simulate the unsatis�eddemand (the calls which encounter busy signals for which we have no data since theblocking occurs in the public telephone network), in the simulation, the tra�c was fedinto a system running with 202 lines, of which 18 were allocated to the �rst stage, and184 to the second stage. The call arrival times and bid durations are from the real tra�ctrace, while bid prices are randomly generated from F , the uniform distribution on [0; 1).Since the arrival process is clearly time-varying, the implemented reservation mechanism

makes real-time estimates of �, and r, to evaluate the constants A;K3;D and K4.Figure 2 shows a snapshot of the simulation trace. The �rst plot shows the number of

users arriving in each batch interval, where 1�= 1 minute. As expected, prices (shown in

the second plot) are zero when the system is lightly loaded during the night. The averageprice including days and nights is 0.2055.The third and fourth plots show the number of users in the second stage, in the reserved

and unreserved states. In this simulation, all users accept the reservation o�er, thus theyonly remain in the unreserved state for a brief period, from the time the o�er is sent by

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0 1 2 3 40

10

20

days

Arriv

als

per b

atch

acis.trace

0 1 2 3 40

0.5

1

days

pric

e

0 1 2 3 40

10

20

30

days

Unr

eser

ved

0 1 2 3 40

100

200

days

Res

erve

d

0 1 2 3 40

50

100

150

days

valu

e

0 1 2 3 40

2

4

6x 10

5

days

cum

. rev

enue

(MO

U)

Figure 2. Simulation trace

the auctioneer (which is a server program), until the acceptance returns from the user(which is a client program on a di�erent host).In the �fth plot, \value" is the sum of the bid prices in the second stage (reserved+unreserved):

this represents the social welfare, or the total value that the users are getting from thesystem. Without pricing, i.e. if all the users were simply admitted on a FCFS basis, theaverage value during peak periods would be 184� 0:5 = 92, since the average bid price is0:5. Here the average value during the day is approximately 110, i.e. the pricing mecha-nism yields a 20% gain in e�ciency. In other words, without pricing there are many timeswhen a user who values access less is denying a user who values it more. The e�ciencyis shown more clearly in Figure 3, normalized so that the horizontal line corresponds tothe FCFS system. Note that there is an e�ciency gain of up to 10% even during o�-peakhours, because some of the o�-peak users arrived during peak hours.The sixth plot of Figure 2 shows the cumulative revenue from usage charges and reser-

vation fees. Since the batch interval is one minute, and usage charges are assessed on aminutely basis. We take the price of 1 as the \full price" reference level, i.e. one unit ofrevenue is when 1 line is charged a price of 1 for 1 minute. The unit is called an MOU(minute-of-use). As can be seen on the plot, the revenue for one day is approximately125,000 MOUs. The 184 lines multiplied by 1440 minutes in a day yield about 264,960potential minutes of use. Thus, on average, a modem line generates 0.47 MOUs of revenueper minute, including usage and reservation charges.Figures 4 and 5 show how the reservation fee o�sets any attempt to \arbitrage". Dura-

tions range from 0 to 24 hours, and the reservation fees range from 0 to 640 MOUs, with

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0 0.5 1 1.5 2 2.5 3 3.5 40.8

0.9

1

1.1

1.2

1.3

1.4

1.5

days

effic

ienc

y

Figure 3. E�ciency: 1.0 = �rst-come �rst-served system

the highest ratio being 1 MOU of reservation fee per minute of reservation duration, ascan be seen by the straight line upper-bounding the scatter plot in Figure 4. The highestratio (most expensive reservation) would be for a user who gets into the system with abid price near zero and requests a long reservation at a time when the arrival rate is high.That is illustrated by Figure 5. A user asking for a long duration reservation at a lowbid price must pay a higher fee which compensates for the expected future rise in prices.The vertical variations are due to the market price at the moment the reservation o�er ismade, where the higher end are reservations made when the market price is high.

REFERENCES

1. A. A. Borovkov. Stochastic Processes in Queueing Theory. Springer-Verlag, 1976.2. A. Das and R. Srikant. Di�usion approximations for models of congestion control in

high-speed networks. In Proc. IEEE Conf. Decision and Control, 1998.3. J. C. Hull. Options, Futures and Other Derivatives. Prentice-Hall, 1997.4. D. L. Iglehart. Limiting di�usion approximations for the many server queue and the

repairman problem. J. Appl. Prob., 2:429{441, 1965.5. I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer-

Verlag, 1991.6. A. A. Lazar and N. Semret. Design and analysis of the progressive second price

auction for network bandwidth sharing. Telecommunication Systems { Special issueon Network Economics, 1999. Available as Tech. Rep. CU/CTR/TR 497-98-21.

7. N. Semret. Market Mechanisms for Network Resource Sharing. PhD thesis, ColumbiaUniversity, 1999.

8. W. Whitt. On the heavy-tra�c limit theorem for GI/G/1 queues. Adv. Appl. Prob.,14:171{190, 1982.

9. W. Whitt. Heavy-tra�c approximations for service systems with blocking. AT&TTech. J., 63(5):689{708, 1984.

10

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

102

reservation duration (min)

rese

rvat

ion

fee

(MO

U)

Figure 4. The reservation fee is at most 1� duration.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

bid price

resv

_fee

/dur

atio

n

Figure 5. Reservations are proportionally more expensive for low bid prices.