Consider a finite-capacity telecommunications link to which connection requests arrive in a Poisson process. Each connection carried on the link earns a certain amount of revenue for the link's manager. Now, assume that the manager is offered the opportunity to buy or sell a unit of the link's allocated capacity. Assuming that the manager has a knowledge of the current number of connections on the link, we demonstrate a method of calculating the buying and selling prices.B.A. Chiera and P.G. Taylo

Chiera, Belinda A.

Taylor, Peter Gerrard

Probability in the Engineering and Informational Sciences

B.A. Chiera

P.G. Taylor

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WHAT IS A UNIT OF CAPACITY WORTH?

Adelaide Research & Scholarship

 PUBLISHED VERSION       Chiera, Belinda A.; Taylor, Peter Gerrard.  What is a unit of capacity worth?, Probability in the Engineering and Informational Sciences, 2002; 16 (4):513-522. Copyright © 2002 Cambridge University Press                 http://hdl.handle.net/2440/611  PERMISSIONS http://journals.cambridge.org/action/stream?pageId=4088&level=2#4408  The right to post the definitive version of the contribution as published at Cambridge Journals Online (in PDF or HTML form) in the Institutional Repository of the institution in which they worked at the time the paper was first submitted, or (for appropriate journals) in PubMed Central or UK PubMed Central, no sooner than one year after first publication of the paper in the journal, subject to file availability and provided the posting includes a prominent statement of the full bibliographical details, a copyright notice in the name of the copyright holder (Cambridge University Press or the sponsoring Society, as appropriate), and a link to the online edition of the journal at Cambridge Journals Online.  Inclusion of this definitive version after one year in Institutional Repositories outside of the institution in which the contributor worked at the time the paper was first submitted will be subject to the additional permission of Cambridge University Press (not to be unreasonably withheld).  6th  May 2011  WHAT IS A UNITOF CAPACITY WORTH?B. A. CHIERA AND P. G. TAYLORDepartment of Applied MathematicsUniversity of AdelaideAdelaide, South AustraliaE-mail: {bchiera,ptaylor}@maths.adelaide.edu.auConsider a finite-capacity telecommunications link to which connection requestsarrive in a Poisson process+ Each connection carried on the link earns a certainamount of revenue for the link’s manager+ Now, assume that the manager is offeredthe opportunity to buy or sell a unit of the link’s allocated capacity+ Assuming thatthe manager has a knowledge of the current number of connections on the link, wedemonstrate a method of calculating the buying and selling prices+1. INTRODUCTIONThere are many situations in which we might wish to place a value on the amount ofcapacity that is allocated to a telecommunications link+ An obvious case occurswhen the manager of the link has the option to purchase extra capacity, which willallow more customers to be carried, or to sell capacity, which will restrict the num-ber of customers that can be carried+The question arises as to how much the managershould pay or charge for the resources that can be bought or sold+Perhaps less obviously, the issue of the value of capacity is crucial to a class ofschemes which can be used for the management of complex telecommunicationsnetworks+ Consider the situation where origin–destination pairs are allocated logicalpaths ~such as virtual paths in ATM networks or label-switched paths in multipro-tocol label switching, MPLS, networks! that share physical resources+ The numberof connections that can be carried on such a logical path depends on some ~perhapsnotional! amount of capacity that is allocated to the path+ If logical paths are allowedto transfer capacity between themselves, then we have precisely a situation where itmakes sense to allocate a value to capacity+ Even though logical paths may not beProbability in the Engineering and Informational Sciences, 16, 2002, 513–522+ Printed in the U+S+A+© 2002 Cambridge University Press 0269-9648002 $12+50 513purchasing capacity in any real sense, efficient management schemes can be de-signed by assuming that they operate as if they are in such a “capacity market+”In this short note, we consider the problem of placing a value on capacity in anM0M0C0C loss system; that is, a C-server loss system where each of the C serversrepresents the amount of capacity needed to carry one connection on a link+ The linkcan buy or sell capacity so that C is increased to C 1 1 or decreased to C 2 1+Previous work in this area, including that of Lanning, Massey, Rider, and Wang@6# and the references therein, has studied the prices that should be charged to cus-tomers in a dynamic loss system+ This work is the closest in spirit to ours+ However,the perspective of @6# is that of an Internet billing system where arrival rates are usercost dependent+ Fulp and Reeves @3# concentrated more heavily on multimarketscenarios, where the price of bandwidth is determined on the basis of current andfuture usage+Other proposed models include WALRAS ~see Wellman @7# ! which computesprices for bandwidth trading in a market-oriented environment by use of the taton-nement process in a competitive equilibrium+ This model was set up as a producer–consumer problem which requires the simultaneous solution of three constrainedlinear programming ~LP! problems+ WALRAS has since been used in later research,in particular by Yamaki, Wellman, and Ishida @8# , where bandwidth is traded atprices computed to reflect current and future capacity requirements+ The main dis-advantage of the WALRAS model is that the time for computation can exceed thetime in which the underlying market changes+ This problem was overcome, to somedegree, in Yamaki, Yamaauchi, and Ishida @9# , where the authors demonstrated thatit is possible to interrupt the WALRAS calculation after some “adequate” time spanand use the interrupted prices+ The model is then restarted with the updated networkinformation, only to be interrupted at some further “adequate” point in time+In this article,we present an alternative approach to the pricing of capacity+First,we present a scheme which, given certain network parameters, will compute the ex-pected lost revenue due to blocking, conditional on the system starting in a given state+We then translate this expected lost revenue into both a buying price and selling pricefor capacity in the system, again relying on knowledge of the state at time zero+The article is organized as follows+ In Section 2, we define a model to computethe expected lost revenue based on the current state of the system+ In Section 3, amethod for the translation of the expected loss in revenue into buying and sellingprices is given+ Our conclusions are given in Section 4+2. MODEL AND ANALYSISAs is well known ~see, e+g+, @5# !, the M0M0C0C loss system can be modeled by acontinuous-time Markov chain with state space $0,1, + + + ,C% and transition rates ofthe formqn, n11 5 Hl, 0 # n , C0, n 5 C, (1)qn, n21 5 nµ, n # C, (2)514 B. A. Chiera and P. G. Taylorwhere l and µ21 are positive constants denoting the arrival rate and mean holdingtime of connections, respectively+ Arrivals to a fully occupied system are lost+ Let udenote the expected revenue generated per accepted arrival to the system+ We set upa model to compute the expected loss in revenue, conditional on a knowledge of thecurrent number of customers in the system, as follows+Let Rn~t ! denote the expected revenue lost in the interval @0, t # given that thereare n connections at time 0 and let Rn~t 6x! be the same quantity conditional on thefact that the first time that the network departs from state n is x+ Because the linkloses revenue at rate ul when C connections are present and not at all when less thanC connections are present, we see thatRn~t 6x! 5 50, n , C, t , xult, n 5 C, t , xnµl 1 nµRn21~t 2 x! 1ll 1 nµRn11~t 2 x!, n , C, t $ xulx 1 RC21~t 2 x!, n 5 C, t $ x+(3)When there are n connections on the link, let Fn be the distribution of the time untilthe first transition+ Then,Rn~t ! 5E0‘Rn~t 6x! dFn~x!+ (4)Due to the Markovian nature of the model, Fn is exponential with parameter l 1 nµwhen n , C and exponential with parameter Cµ when the link is full+ Substituting ~3!into ~4!, we see that there are three cases that we need to consider:Case 1: n 5 0+ In this case, dF0~x! 5 le2lx dx+ From ~3!, we haveR0~t ! 5E0tR0~t 6x! dF0~x!5E0tR1~t 2 x!le2lx dx+ (5)Case 2: 0 , n , C+ In this case, dFn~x! 5 ~l 1 nµ!e2~l1nµ!x dx, and from ~3!, Rn~t !becomesRn~t ! 5E0tRn~t 6x! dFn~x!5E0t@nµRn21~t 2 x! 1 lRn11~t 2 x!#e2~l1nµ!x dx+ (6)WHAT IS A UNIT OF CAPACITY WORTH? 515Case 3: n 5 C+ Finally, in the third case, dFC ~x! 5 ~Cµ!e2~Cµ!x dx andRC ~t ! 5E0tRC ~t 6x! dFC ~x! 1Et‘RC ~t 6x! dFC ~x!5 CµE0t@RC21~t 2 x! 1 ulx#e2~Cµ!x dx1 CµEt‘ulte2~Cµ!x dx,which is equivalent toRC ~t ! 5 CµE0tRC21~t 2 x!e2Cµx dx 1ulCµ~1 2 e2Cµt !+ (7)Taking the Laplace transforms of ~5!–~7!, we see that ERn~s! 5 *0‘e2stRn~t ! dt sat-isfies the system of equationsER0~s! 5ls 1 lER1~s!, (8)ERn~s! 5ls 1 nµ 1 lERn11~s! 1nµs 1 nµ 1 lERn21~s!, 0 , n , C, (9)ERC ~s! 51s 1 Cµ SCµ ERC21~s! 1 uls D+ (10)For network parameters C, l, µ, and u, the solution and inversion of ~8!–~10! willgive the expected lost revenue in @0, t # conditional on there being n connections attime 0+To solve these equations, note that ~8! and ~9! can be written in the formPn11~j! 5 ~j 2 ~dn 1 f !!Pn~j! 2 n~gn 1 h!Pn21~j!, n $ 0, (11)where d 5 2µ0l, f 5 21, g 5 0, h 5 µ0l, and j 5 s0l+ The recurrence relation ~11!describes the class of Meixner polynomials @2, p+ 164# + If, as is the case here, therecurrence relation can further be written asPn11~j! 5 ~j 2 d~n 1 hd22 !!Pn~j! 2 hnPn21~j!,then the solution is known to bePn~j! 5 d nCn~a!S jd D, (12)where a 5 hd22 and Cn~a!~{! is a Charlier polynomial+ Charlier polynomials can begenerally expressed in terms of Laguerre polynomials via the relationCn~a!~j! 5 n!Ln~j2n!~a!+516 B. A. Chiera and P. G. TaylorHence,Pn~j! 5 d nn!Ln~j0d2n!~a!+ (13)It follows that the solution to ~8! and ~9! isERn~s! 5 A~s!PnS slD, (14)where A~s! is a function of s and Pn is given by ~13!+ To find A~s!, we use ~10!, whichgives usERC ~s! 5 A~s!PCS slD5 1s 1 Cµ SCµ A~s!PC21S slD1 uls Dand, therefore,A~s! 5 S 1sDS ul~s 1 Cµ!PC ~s0l! 2 CµPC21~s0l!D+Thus, the general solution for ERn~s! can be written asERn~s! 5 S 1sDS ul~s 1 Cµ!PC ~s0l! 2 CµPC21~s0l!DPnS slD+ (15)In order to derive Rn~t !, we need to invert ERn~s!+ In our numerical investigationspresented below, we use the Euler method outlined in Abate and Whitt @1# + However,there are a number of methods available in the literature+ For a description of thesealternate methods, the reader is referred to Abate and Whitt @1# and Garbow, Giunta,and Lyness @4# +We now present an example of the computation of Rn~t ! for a small system withC 5 6+Example 1: M0M0606 Loss System+ Let C 5 6, u 5 1, l 5 3, and µ21 5 0+5+ Thevalues of Rn~t ! for n 5 $0, + + + ,6% and t [ @0,10! are given in Figure 1+ The functionR0~t ! is the lowest curve and the function R6~t ! is the highest curve+As we would expect, we observe that, for all t, Rn~t ! increases with n+ We alsoobserve that, with increasing t, Rn~t ! is well approximated by a linear function witha slope that is independent of n+ This slope is, in fact, equal to ulE~1+5,6!, whereErlang’s functionE~ r,C! [rC0c!(i50Cr i0i!(16)gives the equilibrium probability that the link is full when the traffic is r 5 l0µ andthe capacity is C+ This observation makes sense, because ulE~1+5,6! is the equilib-rium rate of losing revenue+ The difference in the height of the linear part of theWHAT IS A UNIT OF CAPACITY WORTH? 517functions Rn11~t ! and Rn~t ! reflects the difference in expected lost revenue incurredbefore equilibrium is reached when the system starts with n 1 1 customers ratherthan n customers+In Figure 1, we presented the lost revenue functions for a system with lowblocking ~E~1+5,6! 5 0+00353!+ Figure 2 gives the lost revenue functions for a sys-tem with high blocking+ This has been achieved by increasing the mean holding timeµ21 to 2+ The blocking probability E~6,6! is equal to 0+26492+The traffic, and hence the equilibrium slope of the curves, is much greater inFigure 2 than in Figure 1+ However, the latter is still given by ulE~ r,C!+ The dif-Figure 1. Lost revenue functions for n50, + + + ,6 when C56, l53, and µ21 50+5+Figure 2. Lost revenue functions for n 5 0, + + + ,6 when C 5 6, l 5 3, and µ21 5 2+518 B. A. Chiera and P. G. Taylorference in the equilibrium height of the functions Rn11~t ! and Rn~t ! does not vary asmuch between n 5 0 and n 5 5 as for the low blocking system+ This reflects the factthat, in the low blocking system, states with high occupancy are unlikely to be vis-ited in the short term if the link does not start with a high occupancy+ Thus, thepenalty associated with starting in states with a high occupancy is high compared tothe penalty associated with starting in states with low occupancy+ In the high block-ing system, the probability of moving to states with high occupancy in the short termis relatively higher even if the starting state has a low occupancy+Example 2: M0M01000100 Loss System+ In this example, we consider a larger sys-tem with parameters C 5100, u51, l5 85, and µ21 51+ The values of Rn~t ! for n [$0,25,50,75,90,100% and for t [ @0,10! are shown in Figure 3+As with the smaller loss system, we immediately observe that after an initialperiod in which the starting state has an effect, the Rn~t ! increase linearly at the samerate+ They are also increasing in n, with more pronounced increases as n becomeslarge+3. SETTING THE PRICE OF BANDWIDTHHaving determined the expected lost revenue in the time interval @0, t # , we are leftwith the problem of converting this into prices at which one unit of extra capacityshould be “bought” or “sold+” Let us assume that the network manager is makingbuying and selling decisions for a planning horizon of T time units+ The selection ofT is a decision for the network manager+ If the link is such that opportunities forcapacity trading occur every u time units, one possibility would be to make T 5 u+Such a choice would be myopic in the sense that it does not take into account theFigure 3. Lost revenue functions for n 5 0, 25, 50, 75, 90, and 100 when C 5100,l 5 85, and µ21 5 1+WHAT IS A UNIT OF CAPACITY WORTH? 519number of connections on the link at the end of the time period nor, indeed, theresidual value of having extra capacity at time u+Once the manager has chosen T, we can regard the value of an extra unit ofcapacity as the difference in the total expected lost revenue over time @0,T # if thesystem were to increase its capacity by one unit at time zero+ Conversely, we cancalculate the selling price of a unit of capacity as the difference in the total expectedlost revenue over time @0,T # if the system were to decrease its capacity by one unit+The buying and selling prices, Bn~T ! and Sn~T !, respectively, of bandwidthwhen the initial state is n and the planning horizon is T can thus be written asBn~T ! 5 Rn,C ~T ! 2 Rn,C11~T !, (17)Sn~T ! 5 Rn,C21~T ! 2 Rn,C ~T ! (18)where the extra subscript in Rn,C ~T ! indicates the initial capacity+We expect that, forall n and T, Sn~T ! . Bn~T !+We give some examples of the computation of Bn~T ! andSn~T !+Example 3: M0M0606 Loss System+ For the low blocking system with C 5 6, u 51,l 5 3, and µ21 5 0+5, the buying and selling prices of bandwidth, Bn~T ! ~dottedlines! and Sn~T ! ~continuous lines! for n54, 5, 6 are displayed in Figure 4+The samedata for the high blocking system with C 5 6, u 51, l 5 3, and µ21 5 2 is shown inFigure 5+From Figures 4 and 5, we immediately observe that Sn~T ! is greater than Bn~T !for all n and T in both systems+ We see that as n approaches link capacity C, thesystem places an increasingly higher value on the available capacity, both frombuying and selling points of view+ Also, both Sn~T ! and Bn~T ! are steeper for theFigure 4. Buying and selling price functions for n 5 4, 5, 6 when C 5 6, l5 3, andµ21 5 0+5+520 B. A. Chiera and P. G. Taylorlink with greater load, but the distances Sn11~T !2Sn~T ! and Bn11~T !2Bn~T ! showless variation as n varies+As with the lost revenue functions, it is useful to note that for large T, Sn~T ! andBn~T ! are well approximated by linear functions+ More precisely Sn~T ! can be ap-proximated by a function of the form ZSn~T ! 5 sn 1 ul~E~ r,7! 2 E~ r,6!!T andBn~T ! can be approximated by a function of the form ZBn~T ! 5 bn 1 ul~E~ r,6! 2E~ r,5!!T+ The values of sn and bn reflect the total contributions to the buying andselling prices of capacity that are accumulated when the system is in its transientstage+Example 4: M0M01000100 Loss System+ The values of Bn~T ! and Sn~T ! for n [$50,75,90,100% are given in Figure 6+ Similar observations can be made as for thesmaller system+ The buying prices are always lower than the selling prices, and bothincrease markedly as the link nears full occupancy+4. CONCLUSIONSIn this article, we have presented a model which, given the current state of a C-serverloss system, computes the expected loss in revenue due to blocking+ Assuming agiven planning horizon, we then translated these expected losses into buying andselling prices of one unit of bandwidth+ The main motivation for considering themodel presented in this article is that it could be implemented in future work as anintegral part of a network management scheme in which logical paths trade capacitybetween themselves+We formulated a model on the basis of a system of renewal equations and thenderived a system of recurrence relations satisfied by the Laplace transform of Rn~t !+It was shown that this system of recurrence relations was of Meixner type, for whichFigure 5. Buying and selling price functions for n 5 4, 5, 6 when C 5 6, l5 3, andµ21 5 2+WHAT IS A UNIT OF CAPACITY WORTH? 521a solution could be determined in terms of Laguerre polynomials+ We inverted theseLaplace transforms numerically using the Euler method+We demonstrated the computation of these prices for both a small system withC 5 6 and a more realistically sized system with C 5 100+AcknowledgmentThe authors would like to thank the Australian Research Council for supporting this work through grantA10033153+References1+ Abate, J+ & Whitt, W+ ~1995!+ Numerical inversion of Laplace transforms of probability distributions+ORSA Journal on Computing 7: 36– 43+2+ Chihara, T+ ~1978!+ An introduction to orthogonal polynomials+ New York: Gordon and Breach+3+ Fulp, E+ & Reeves, D+ ~2000!+ Qos rewards and risks:A multi-market approach to resource allocation+In Proceedings of the IFIP-TC6 Networking 2000 Conference, pp+ 945–956+4+ Garbow, B+, Giunta, G+, & Lyness, J+ ~1988!+ Software for an implementation of weeks’method for theinverse Laplace transform problem+ ACM Transactions on Mathematical Software 14~2!: 163–170+5+ Kleinrock, L+ ~1975!+ Queuing systems+ Vol+ 1: Theory+ New York: Wiley+6+ Lanning, S+, Massey, W+, Rider, B+, & Wang, Q+ ~1999!+ Optimal pricing in queueing systems withquality of service constraints+ In Proceedings of the 16th International Teletraffic Congress (ITC 16),pp+ 747–756+7+ Wellman, M+ ~1993!+ A market-oriented programming environment and its application to distributedmulticommodity flow problems+ Journal of Artificial Intelligence Research 1: 1–23+8+ Yamaki, H+, Wellman, M+, & Ishida, T+ ~1996!+ A market-based approach to allocating qos formultimedia applications+ In Second International Conference of Multi-Agent Systems (ICMAS-96),pp+ 385–392+9+ Yamaki, H+, Yamaauchi, Y+, & Ishida, T+ ~1998!+ Implementation issues on market-based qos control+In International Conference on Multi-Agent Systems (ICMAS-98), pp+ 357–364+Figure 6. Buying and selling price functions for n550, 75, 90, 100 when C5100,l 5 85, and µ21 5 1+522 B. A. Chiera and P. G. Taylor

What is a unit of capacity worth?

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What is a unit of capacity worth?

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