MANAGEMENT SCIENCE doi /mnsc ec pp. ec1 ec12

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1 MANAGEMENT SCIENCE doi 0.87/mnsc ec pp. ec ec e-companion ONLY AVAILABLE IN ELECTRONIC FORM informs 008 INFORMS Electronic Companion Service Adoption and Pricing of Content Delivery Network (CDN) Services by Kartik Hosanagar, John Chuang, Ramayya Krishnan, and Michael D. Smith, Management Science, doi 0.87/mnsc

2 Online Appendix for Service Adoption and Pricing of Content Delivery Network (CDN) Services Kartik Hosanagar, John Chuang, Ramayya Krishnan, Michael D. Smith Appendix A We can use the conjugate pairs theorem from calculus (Currier 000) to analyze the properties of the optimal infrastructure level, I * ( λ). The theorem states that for the problem max F( x, a), the derivative x * x / a and the cross partial F xa have the same sign. The following results follow by applying the theorem: i) If the cost of infrastructure increases, I * decreases. * I U Ia = -. This implies that < 0. a As expected, if the infrastructure costs (cost of processing and bandwidth) decrease, the optimal level of investment in infrastructure increases. ii) If there are significant economies in scale in content delivery, I * increases. * I U Ib = I > 0. Thus > 0. b That is, if server or bandwidth sellers provide high volume discounts, infrastructure levels of CPs will increase. iii) If a content provider s cost of losing requests is high, I * is correspondingly higher. K ( K + ) λ ( I λ) I λ Proof: U Ic =. We know from the first order condition that ( I λ ) I λ - c λ c ( K + ) λ ( I λ) I {a - b I}- + I λ ( I λ ) K = 0. Since -{a - b I} < 0 (the rate at which infrastructure costs increase with infrastructure level I), it follows that c λ + K I λ c ( K + ) λ ( I λ) I + ( I λ ) - K + K = cu Ic > 0. Since c > 0, it follows immediately that

3 * I U Ic > 0. From the conjugate pairs theorem, > 0. c In other words, if the cost (c) of losing a request increases, the optimal infrastructure level also increases. iv) If the arrival rate of requests λ increases, I * increases. Proof: This statement follows from conjugate pairs theorem if U > 0 is true. Computing the cross partial with respect to I, λ and simplifying, K K { K( I λ)( I + λ ) Iλ( I λ )}. K K c( K + ) λ I U I λ = Thus, U 3 Iλ > 0 if and only if ( I λ ) Iλ K ( I λ)( I + λ ) Iλ( I K K λ ) > 0. This can be restated as K + K + U Iλ > 0 if and only if Kp ( K + ) p + ( K + ) p K > 0 (OA) where p = I / λ. For I >> λ, it follows that p >>. Thus ( K + ) p > K (OA) Also, for large K (queue size), we know the following is true: p > +. This can be restated as: K K + K + Kp > ( K + ) p (OA3) Adding (B) and (C) yields Kp K + + ( K + ) p > ( K + ) p + K. Combining this result with (A), it K + * I follows that U Iλ > 0. That is, > 0. λ QED. Appendix B: Numerical Results I. Numerical Results on CP s Infrastructure Sizing (Poisson Traffic) We employ numerical tests using parameter values conforming to typical bandwidth and hosting costs to determine the approximate relationship between λ and I *. The CP s optimal infrastructure is

4 determined by trading off the cost of infrastructure against the cost of lost requests. We now fix the parameters associated with these two costs. For the infrastructure cost function, C( I) I = a I a, we assume that a =3.56 and a = These values roughly correspond to current infrastructure costs. For example, under these parameter values, the cost of serving 33 requests/min is $804 per month. If we assume that the average size of the response to a request is 50 Kbytes, this implies that the cost of serving data at.55 Mbps is $804 per month. This is reasonable given the cost of a T connection (approximately $400 per month) and maintaining a workstation. Likewise, the cost of serving 6,975 requests per minute is $,04, which is also approximately the cost of a T3 connection and the associated cost of maintaining a server. Finally, the cost of serving 3,55 requests per minute is $57,08 per month, roughly equivalent to the cost of an OC3 connection. These costs are also comparable to managed hosting costs at the time of this study. We assume that the cost of a lost request, c, is $0. This is based on an assumption that 0% of visitors purchase products/services with an average purchase of $00, and a customer leaves a website if a request does not go through. Finally, we assume that the queue size, K, (for requests waiting to be processed) is 04 requests. Figure A shows the optimal infrastructure level (in requests/min) for different arrival rates ranging from 5,000 to 0,000 requests per minute. The relationship is approximately linear. To test for robustness, we repeated the numerical analysis for a variety of other settings for buffer size K and cost of lost requests c, and found that the relationship is approximately linear in all cases. For example, Figure A shows the relationship for the case where { a = 3.46; a = ; K = 56; c = * that the special case where I = λ (boundary solution) is also linear. 0}. While the plotted solutions are all interior solutions, note

5 Optimal Infrastructure Level Request Arrival Rate Optimal Infrastructure Level Request Arrival Rate Figure OA: Optimal Infrastructure Level versus Arrival Rate (Case ) Figure OA: Optimal Infrastructure Level versus Arrival Rate (Case ) Case of Multiple Servers: We relax the assumption of a single server system and numerically evaluate the CP s capacity sizing decision for a three server system. The queue size is 4, cost of lost requests is $0 and all other settings are the same as in Figures OA and OA. The optimal infrastructure level for different arrival rates is plotted in Figure OA3. The relationship continues to be approximately linear. As is intuitive, the optimal infrastructure level for each server (I * ) can now be lower than the mean arrival rate, λ, as three servers are sharing the load. Optimal Infrastructure Level Request Arrival Rate Figure OA3: Optimal Infrastructure Level Vs. Arrival Rate (with Three Servers)

6 Sensitivity to Cost Parameters: We test the impact of different cost parameters on the CP s optimal infrastructure level. Figure OA4 shows the relationship for the case where the cost of a lost request is scaled up to c=$80. All other parameters are the same as in Figure OA, i.e., a = 3.46; a = ; K 56}. In Figure OA5, { = we test the impact of a decline in the CP s infrastructure costs. Specifically, we set a =.5 and leave the remaining parameters unchanged Optimal Infrastructure Mean Arrival Rate Figure OA4: Optimal Infrastructure Level Vs Arrival Rate (Cost of lost requests = $80) Optimal Infrastructure Mean Arrival Rate Figure OA5: Optimal Infrastructure Level Vs Arrival Rate (a =.5) As expected, the infrastructure levels are higher than in the base case. That is, if the cost of lost requests increases or the infrastructure cost decreases, the optimal infrastructure increases. In both these

7 instances, the nearly linear relationship between mean arrival rate and optimal infrastructure is maintained. Thus the numerical analysis supports the assumption in Section 3.. (optimal infrastructure varies approximately linearly with mean demand). II. Numerical Results on Optimal Infrastructure Sizing (MMPP Traffic) We now study optimal infrastructure sizing under bursty traffic. All parameters except the traffic at the CP are the same as in the previous subsection (Figure OA). This includes cost of infrastructure, cost of a lost request and buffer size. For the traffic at the CP server, we assume MMPP parameters as described in section 3.. ( λ = 0λ ; ( q = 0.9, q = 0.) ). In addition, we maintain constant burstiness ( Ψ / λ ) across CPs. Figure OA6: Optimal Infrastructure Level for Poisson and MMPP traffic Figure OA7: Net cost incurred by CP The number of lost requests associated with any choice of I can be computed numerically and the optimal infrastructure level can be computed by trading off the infrastructure cost and cost of lost requests. In Figure OA6, we present the optimal infrastructure level associated with a given mean arrival rate under the assumption of MMPP traffic. For the sake of comparison, we also present the infrastructure levels if the traffic were Poisson (from Figure OA). Interestingly, we find that the CP s optimal infrastructure level with bursty traffic is lower than that with Poisson traffic, even though MMPP traffic has higher variance than Poisson traffic for the same mean arrival rate. This is driven by the fact that most of the lost

8 requests are due to bursts that occur when the system in state. In order to see a marked reduction in lost requests, the infrastructure level has to be raised above λ. However, this raises the infrastructure cost substantially and is suboptimal. Thus, the high disparity in arrival rates between the two states implies that the CP is often willing to accept server downtime during bursts, and thus the optimal infrastructure level is driven by λ and not λ. Since λ < λ, the optimal infrastructure level is also lower than that with Poisson traffic. In Figure OA7, we plot the CPs net cost (infrastructure cost + cost of lost requests) and observe that it is higher with MMPP traffic than under Poisson traffic. Thus, the CPs expected surplus is lower when the traffic is MMPP. The primary takeaway from this section is that the approximately linear relationship between optimal infrastructure and mean arrival rate is valid under bursty traffic as well. III. Numerical Results on Pricing with Bursty Traffic In this section, we determine the impact of cost and traffic parameters on the optimal price and resulting profits of the CDN. We first test the impact of a decrease in the CPs infrastructure costs. With declining bandwidth costs and costs of managed hosting solutions, this is particularly relevant. In the second row of Table OA, we present the simulation results on optimal usage-based prices for the case where a =.5. The results for the base case (a =3.46) from the paper are also presented for comparison (row ). When a CP s infrastructure costs decline, CP surplus from self-provisioning increases, and hence CDNs have to lower their price to continue to attract customers. This is consistent with the analytical results from Section 3.. We also test the impact of a decline in the CDN s variable cost by setting b =. The results are presented in row 3 of Table OA. Consistent with the analytical results of Section 3., the optimal price decreases when the CDN s costs decrease. Thus in reality, when infrastructure costs decrease, prices will decrease substantially, driven by both the increased CP surplus from selfprovisioning and the decrease in cost of operating a CDN. Interestingly, in all these cases, we observe that If the CP s cost is convex in the number of lost requests, then the CP may be more willing to incur higher infrastructure costs in return for lower cost of lost requests. The resulting infrastructure level can be much higher than suggested in Figure C. Regardless, bursts will continue to negatively impact the CP s surplus.

9 the optimal volume-based price function entails volume discounts when traffic is Poisson or pure MMPP (MMPP with same burstiness across CPs). However, CPs are charged a volume tax when the traffic is mixed, i.e. characterized by heterogeneity in burstiness across CPs. Poisson MMPP Mixed { a =3.46, b =3} 3.9X 6.6e 8.5X 4.9e 4.8X + 7.6e { a =.5, b =3} 3.0X 5.6e { a =3.46, b =} 3.4X 6.6e 7.X 6.6e 8.X 5.6e 3.7X + 7.6e 4.X + 5.6e Table OA: Optimal Price Functions for the Three Cases In each of the two new cases (first with a =.5 and second with b =), we also determine the CDN profits under pricing at the 95 th percentile of usage. Figures OA8 and OA9 demonstrate that while percentile-pricing provides no significant advantages when traffic is Poisson or pure-mmpp, it can enable a significant increase in profits when the traffic profile is mixed. This supports the results in our paper..60e+07.40e+07.0e+07 CDN Profit.00E E E+06 Usage-based Price 95th Percentile-based Price 4.00E+06.00E E+00 Poisson MMPP Mixed Traffic Profile Figure OA8: CDN Profit with Different Pricing Policies and Traffic Profiles (a =.5)

10 .50E+07.00E+07 CDN Profit.50E+07.00E+07 Usage-based Price 95th Percentile-based Price 5.00E E+00 Poisson MMPP Mixed Traffic Profile Figure OA9. CDN Profit with Different Pricing Policies and Traffic Profiles (b =) IV. Impact of Correlation in Traffic across CPs The simulation results presented in the paper are based on an assumption of independent traffic across CPs. This is generally reasonable given the considerable variety in types of subscribers that CDNs have today. However, it is interesting to consider how correlation in traffic across CPs might impact the CDN s pricing policy. In this subsection, we present the results of our pricing simulations for mixed traffic, assuming a correlation in traffic for the,000 CPs. All the off-diagonal entries in the correlation matrix are set to the same value. Because the primary impact of any correlation in bursts across CPs will be to overwhelm the CDN s infrastructure, we will make assumptions about the CDN s available capacity. If the CDN has infinite capacity, correlations in bursts do not impact the CDN s ability to provide services to the CPs. Thus, correlations in bursts would not impact either the value that CPs get from the CDN or the CDN s variable costs. In the presence of capacity constraints, simultaneous bursts across CPs can overwhelm the CDN s infrastructure and hinder its ability to provide service. We set the CDN s available capacity at 5M requests/period. Further, we assume that the CDN operates under the constraint that the 95 th percentile of the traffic from subscribers should be less than

11 75% of available capacity. Though the specific numbers chosen here are arbitrary, the structure of this decision rule reflects a common rule of thumb employed at CDNs. The simulation results are presented in Figure OA0. In our simulations, we find that initially there is no marked difference in the CDN s optimal price as the correlation coefficient increases. This is because the 95 th percentile of traffic is still within the limit. That is, the optimal price computed is an interior solution. CPs do not face any negative externality from bursts from another CP because any bursts can be easily accommodated by the CDN by rerouting requests and adjusting the number and placement of replicas. However, if the correlation in traffic continues to increase, it also increases the 95 th percentile of the subscriber traffic, eventually resulting in the traffic constraint being met. When this occurs, the CDN is unable to maintain its service and must either increase capacity or limit the traffic in its network. For a given capacity constraint, the CDN must increase its price as shown in Figure OA6. Thus, the net impact of correlation in burstiness is an increase in the price charged by the CDN Price(X=000) Correlation Coefficient Figure OA0. CDN s Price Vs. Correlation in CPs traffic Appendix C: CDN Simulations The simulations discussed in this Section are implemented on the CDNSim simulation platform. CDNSim is a trace-driven CDN simulator that simulates a network with clients, CDN servers and origin That is, the specific choice of 75% capacity utilization at the 95 th percentile of traffic is arbitrary, but it reflects a common X% capacity utilization at 95/99 th percentile of traffic rule of thumb.

12 servers interconnected through several intermediate routers. The simulator creates a network once the number of CDN servers and clients is specified. The simulator offers several network topology generators. We use the GT-ITM topology generator (Zegura et al 996) in our simulations. The topology generator creates edges connecting the CDN servers and routers based on the distance between them. CDNSim implements several features of TCP/IP, including packet switching, retransmission, etc. Each client is assigned to one of the CDN servers based on distance. The speed of the connection between the nodes is based on typical bandwidth in the real-world. Requests generated by the clients are forwarded to the designated CDN server and the server responds from its local cache or by forwarding the request to another CDN server or the origin server. The data objects and requests are generated using trace generators as described in Paliis et al (005). Initially, all objects are stored in the origin servers and all CDN servers are empty. During this phase, requests are always forwarded by the CDN servers to the origin servers. The initial request data is used as a training dataset for a placement algorithm that then assigns the objects to the CDN servers in order to minimize latency. One object may be stored in more than one CDN server by the placement algorithm. The training data is used only for object placement and thus performance results below do not account for requests in the training data. The response time for a request is the time between the user issuing a request and subsequently receiving the response. Lastly, simulations in CDNSim are considered valid if the number of request denied is zero or negligible relative to the total number of requests issued. Further details regarding the simulation testbed can be obtained from Pallis et al (005). Below, we focus only on the mean response time as a function of CDN network size. The complexity of the objects placement algorithm with N servers in CDNSim is O(N 4 ) and hence we restrict the simulations to a maximum of 50 CDN servers (which takes over a day to run). In Figure OA, we report the mean response time as a function of the number of CDN servers. We observe that the mean response time decreases with the number of CDN servers in a convex manner. Thus, the benefit from the CDN (improvement in response time) increases with the number of CDN servers in a concave fashion. This is consistent with the empirical observations in Cronin et al (00). Thus, we model the mean

13 response time per request as an increasing function of the number of CDN servers. Generally, the simulations suggest that the mean response time is strongly influenced by the network topology and the speed of the connections between the various elements of the networks (clients, CDNs, origin server, routers) Mean Latency Number of CDN Servers Figure OA: Mean Response Time versus Number of CDN Servers References K. Currier. Comparative Statics Analysis in Economics. World Scientific Publishing Company. August 000.