View metadata,citationandsimilarpapersatcore.ac.uk arXiv:1203.2841v2 [cs.NI] 14 Aug 2012 fTcnlg,M S (e-mail: USA MA Technology, of i.e useful application, the and or user usin the experience by goodput of of of perceived number quality quality the users’ users’ the measure reducing the We while stations. that improved base show be base we may more cases, experience installing some without In achieved be stations. may experience of to [2]. able growth be traffic the tha not with indicate may pace trends improvements keep several developme The efficiency as rapid [2]. technology question the such (kW) to kilowatts brought of been 2-3 feasibility have base use the each to and as shown cost sustainability energy been significant has a station at incr this comes However, users. to in experience le new of would quality The which bas higher coverage. bandwidth, to more full more provide have to (e.g. already is service upgrade infrastructure which growth, infrastructure mobile areas and such 1, more in install Figure meet stations) install in to to often shown effort order As providers increasing networks. 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transaction to be ∆ = µf /rg seconds. If the user is already p1 p1+p 2+p 3 p2 p1+2p 2+p 3 active, then the new transaction is added to the user’s queue. p ACK(p ) p +2p +2p seen(p ) 3 Lost 1 1 2 3 Lost 1 If the user has initiated k transactions, the model of adding the ACK(p ) seen(p ) 1 2 jobs into the user’s queue is equivalent to splitting the goodput rg to k transactions (each transaction achieves a rate of rg/k

TCP TCP/NC Mbps). Fig. 2: Example of TCP and TCP/NC. In the case of TCP, the We denote pp to be the probability of in the network, and RTT to be the round-trip time. In a wireless, p TCP sender receives duplicate ACKs for packet p1, which p may wrongly indicate congestion. However, for TCP/NC, the and RTT may vary widely. For example, wireless connection over WiFi may have RTT ranging from tens of milliseconds TCP sender receives ACKs for packets p1 and p2; thus, the TCP sender perceives a longer RTT but does not mistake the to hundreds of milliseconds with loss rates typically ranging loss to be congestion. from 0-10%. In a more managed network (such as cellular networks), RTT are typically higher than that of a WiFi network but lower in loss rates. equations to decode the original data. TCP/NC may not be the only viable solution, and other transport protocols that III. ANALYSIS OF THE NUMBER OF BASE STATIONS can combat erasures may be used. We use TCP/NC for its We analyze n needed to support n users given throughput effectiveness and simplicity. bs r and goodput r . We first analyze P (∆,p), the probability TCP/NC allows a better use of the base stations installed, t g that a user is active at any given point in time. Given P (∆,p), and can improve the goodput without any additional base we compute the expected number of active users at any given stations. Improving the goodput with the same or a fewer point in time and n needed to support these active users. number of base stations implies reduction in energy cost, bs Consider a user u at time t. There are many scenarios in operational expenses, capital expenses, and maintenance cost which u would be active at t. User u may initiate a transaction for the network provider. The results in this paper can also at precisely time t with probability p. Otherwise, u is still in be understood as being able to serve more users or traffic the middle of a transaction initiated previously. growth with the same number of base stations. This may lead To derive P (∆,p), we use the Little’s Law. For a stable to significant cost savings, and may be of interest for further system, the Little’s Law states that the average number of jobs investigation. (or transactions in our case) in the user’s queue is equal to the product of the arrival rate p and the average transaction time ∆. II. MODEL When ∆p ≥ 1, we expect the user’s queue to have on average Consider a network with n users. We assume that these n at least one transaction in the long run. This implies that the users are in an area such that a single base station can cover user is expected to be active at all times. When ∆p < 1, them as shown in Figure 1. If the users are far apart enough we can interpret the result from Little’s Law to represent the that a single base station cannot cover the area, then more probability that a user is active. For example, if ∆p = 0.3, base stations are necessary; however, we do not consider the the user’s queue is expected to have 0.3 transactions at any problem of coverage. given point in time. This can be understood as the user being The network provider’s goal is to provide a fair service to active for 0.3 fraction of the time. Note that when the system is any user that wishes to start a transaction. Here, by fair, we unstable, the long term average number of uncompleted jobs mean that every user is expected to receive the same average in the user’s queue may grow unboundedly. In an unstable throughput, denoted as rt Mbps. The network provider wishes system, we assume that in the long term, a user is active with to have enough network resources, measured in number of probability equal to one. base stations, so that any user that wishes to start a transaction Therefore, we can state the following result for P (∆,p). is able to join the network immediately and achieve an average µf throughput of rt Mbps. We denote rg to be the goodput P (∆,p) = min{1, ∆p} = min 1, · p . (1)  rg  experienced by the user. Note that rg ≤ rt. We denote nbs to be the number of base stations needed to Given P (∆,p), the expected number of active users is meet the network provider’s goal. We assume that every base nP (∆,p). We can now characterize the expected number of station can support at most Rmax Mbps (in throughput) and base stations needed as at most Nmax active users simultaneously. In this paper, we rt 1 assume that Rmax = 300 Mbps and Nmax = 200. nbs = nP (∆,p) · max , . (2)  Rmax Nmax  A user is active if the user is currently downloading a rt 1 file; idle otherwise. A user decides to initiate a transaction In Equation (2), max { Rmax , Nmax } represents the amount with probability p at each time slot. Once a user decides to of base stations’ resources (the maximum load Rmax or the initiate a transaction, a file size of f bits is chosen randomly amount of activity Nmax) each active user consumes. The according to a probability distribution Pf . We denote µf to value of nbs from Equation (2) may be fractional, indicating be the expected file size, and the expected duration of the that actually ⌈nbs⌉ base stations are needed. 3

6 8 6 7

5 5 6 6 5 4 4 4 bs bs bs 3 bs 4 3 n n n n p=0.01 p=0.01 3 p=0.01 p=0.01 2 p=0.02 2 p=0.02 2 p=0.02 2 p=0.02 p=0.03 p=0.03 p=0.03 1 1 p=0.03 p=0.04 p=0.04 p=0.04 1 p=0.04 0 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 5 0 1 2 3 4 5 r (Mbps) r (Mbps) r (Mbps) r (Mbps)

(a) µf = 3.2 MB (b) µf = 5.08 MB (a) µf = 3.2 MB (b) µf = 5.08 MB

Fig. 3: The values of nbs from Equation (2) with n = 1000 Fig. 4: Average value of nbs over 100 iterations with n = 1000 and varying p and r. and varying p and r.

Note the effect of rt and rg . As shown in Equation (2), B. Simulation Results increasing r incurs higher cost while increasing r reduces the t g We present MATLAB simulation results to verify our analy- cost. Therefore, when a network provider dedicates resources sis results in Section IV-A. We assume that at every 0.1 second, to increase r , the goal of the network provider is to increase p t a user may start a new transaction with probability . This r proportional to r . 10 g t was done to give a finer granularity in the simulations; the IV. BEST CASE SCENARIO results from this setup is equivalent to having users start a new transaction with probability p every second. We assume In an ideal scenario, the user should see a goodput r = r . g t that there are n = 1000 users. For each iteration, we simulate In this section, we analyze this best case scnario with r = the network for 1000 seconds. Each plot is averaged over 100 r = r . Once we understand the optimal scenario, we then t g iterations. consider the behavior of TCP and TCP/NC in Section V. Once a user decides to start a transaction, a file size is A. Analytical Results chosen randomly in the following manner. We assume there are four types of files: f = 8KB (a document), f In Figures 3a and 3b, we plot Equation (2) with µf =3.2 doc image = 1MB (an image), f 3 = 3 MB (a mp3 file), f = MB and µf =5.08 MB for varying values of p. As r increases, mp video 20 MB (a small video), and are chosen with probability it does not necessarily lead to increase in nbs. Higher r results in users finishing their transactions faster, which in turn allows pdoc, pimage, pmp3, and pvideo, respectively. In Figure 4a, the resources dedicated to these users to be released to serve we set [pdoc,pimage,pmp3,pvideo] = [0.3, 0.3, 0.3, 0.1]. This other requests or transactions. As a result, counter-intuitively, results in µf = 3.2 MB as in Figure 3a. In Figure 4b, we we may be able to maintain a higher r with the same or a set [pdoc,pimage,pmp3,pvideo] = [0.26, 0.27, 0.27, 0.2], which fewer number of base stations than we would have needed for gives µf =5.08 MB as in Figure 3b. a lower r. For example, in Figure 3a, when r < 1 Mbps, the The simulation results show close concordance to our anal- rate of new requests exceeds the rate at which the requests ysis. Note that the values in Figures 4a and 4b are slightly are handled; resulting in an unstable system. As a result, most greater than that of Figures 3a and 3b. This is because, in the users are active all the time, and the system needs n simulation, we round-up any fractional nbs’s since the number Nmax = 1000 of base stations needs to be integral. 200 =5 base stations. There are many cases where nbs is relatively constant V. ANALYSIS FOR TCP/NC AND TCP regardless of r. For instance, consider p =0.03 in Figure 3b. The value of nbs is approximately 4-5 throughout. However, We now study the effect of TCP and TCP/NC’s behavior. there is a significant difference in the way the resources We use the model and analysis from [5] to model the rela- are used. When r is low, all users have slow connections; tionship between rg and pp for TCP and TCP/NC. We denote therefore, the base stations are fully occupied not in throughput rg−nc to be the goodput when using TCP/NC, and rg−tcp to be but in the number of active users. On the other hand, when r is that for TCP. We set the maximum congestion window, Wmax, high, the base stations are being used at full-capacity in terms of TCP and TCP/NC to be 50 packets (with each packet being of throughput. As a result, although the system requires the 1000 bytes long), and their initial window size to be 1. We same number of base stations, users experience better quality consider RTT = 100 ms and varying pp from 0% to 5%. We of service and users’ requests are completed quickly. note that, given rt and pp, rg ≤ rt(1 − pp) regardless of the When p and r are high enough, it is necessary to increase protocol used. nbs. As demand exceeds the network capacity, it becomes In [5], [10], TCP/NC has been shown to be robust against necessary to add more infrastructure to meet the growth in erasures; thus, allowing it to maintain a high throughput demand. For example, consider p =0.04 in Figure 3b. In this despite random losses. For example, if the network allows case, as r increases nbs increases. for 2 Mbps per user and there is 10% loss rate, then the user 4

4 4 p = 0% p = 0% restricted by Wmax. However, TCP achieves this maximal p p 3.5 p = 1% p = 1% goodput only when pp = 0%. This is because, when there p p p = 2% 3 p = 2% are losses in the network, TCP is unable to recover effectively 3.5 p p p = 3% p = 3% p 2.5 p from the erasures and fails to use the bandwidth dedicated to p = 4% p = 4% p p it. For pp > 0%, rg−tcp is not limited by Wmax but by TCP’s (Mbps) (Mbps) 2 3 p = 5% p = 5% p p performance limitations in lossy wireless networks. g−nc g−tcp r r 1.5

2.5 1 C. The Number of Base Stations for TCP/NC and TCP

0.5 We use the values of rg−nc and rg−tcp from Sections V-A and V-B to compare the number of base stations for TCP/NC 2 0 2 2.5 3 3.5 4 4.5 0 1 2 3 4 r (Mbps) r (Mbps) and TCP using Equation (2). We assume that SRTT = RTT . t t In general, SRTT is slightly larger than RTT . (a) rg−nc (b) rg−tcp Figures 6 and 7 show nbs predicted by Equation (2) when Fig. 5: The value of rg−nc and rg−tcp against rt for varying RTT = 100 ms. TCP suffers performance degradation as values of pp. We set RTT = 100 ms. pp increases; thus, nbs increases rapidly with pp. Note that increasing rt without being able to increase rg leads to should see approximately 2 · (1 − 0.1)=1.8 Mbps. Reference inefficient use of the network, and this is clearly shown by [5] has shown, both analytically and with simulations, that the performance of TCP as rt increases with pp > 0%. TCP/NC indeed is able to achieve goodput close to 1.8 Mbps However, for TCP/NC, nbs does not increase significantly in such a scenario while TCP fails to do so. (if any at all) when pp increases. As discussed in Section III, TCP/NC is able to translate better rt into rg−nc despite − A. Behavior of rg nc with varying pp pp > 0%, i.e. rt ≈ rg−nc. As a result, this leads to a significant Equation (20) from [5] provides the goodput behavior of reduction in nbs for TCP/NC compared to TCP. Note that nbs TCP/NC, which we provide below in Equation (3). for TCP/NC is approximately equal to the values of nbs in 2 Section III regardless of the value of pp. Since TCP/NC is 1 (Wmax − 1) +(Wmax − 1) rg−nc = tWmax − , resilient to losses, the behavior of r − does not change as tSRTT  2  g nc (3) dramatically against pp as that of rg−tcp does. As a result, we where SRTT is the effective RTT observed by TCP/NC observe nbs for TCP/NC to reflect closely the values of nbs and increases with pp and t represents the duration of the seen in Section III, which is the best case with rt = rg. connection (in number of RTTs). Equation (3) shows the effect We observe a similar behavior for other values of RTT of network coding. The goodput of TCP/NC decreases with pp; as we did for RTT = 100 ms. The key effect of the value however, the effect is indirect. As pp increases, the perceived of RTT in the maximal achievable goodput. For example, RTT increases, which leads to TCP/NC reducing its rate. if Wmax is limited to 50, the maximal achievable goodput Combining Equation (3) and rg−nc ≤ rt(1−pp), we obtain is approximately 0.8 Mbps when RTT = 500 ms, which is the values of rg−nc for various rt, RTT , and pp. In Figure much less than the the 4 Mbps achievable with RTT = 100 5a, the values of rg−nc plateaus once rt exceeds some value. ms. As a result, for RTT = 500 ms, neither rg−nc nor rg−tcp This is caused by Wmax. Given Wmax and RTT , TCP/NC and can benefit from the increase in rt beyond 0.8 Mbps. Despite TCP both have a maximal goodput it can achieve. In the case this limitation, TCP/NC still performs better than TCP when with RTT = 100 ms, the maximal goodput is approximately losses occur. When demand exceeds the maximal achievable 4 Mbps. Note that regardless of pp, all TCP/NC flows achieve goodput, nbs increases for both TCP/NC and TCP in the same the maximal achievable rate. This shows that TCP/NC can manner. We do not present the results for want of space. overcome effectively the erasures or errors in the network, VI. CONCLUSIONS and provide a goodput that closely matches the throughput rt. In wireless networks, the solution to higher demand is often B. Behavior of rg−tcp with varying pp to add more infrastructure. This is indeed necessary if all Equation (16) from [5] provides the goodput behavior of the base stations are at capacity (in terms of throughput). TCP, which we provide below in Equation (4). However, in many cases, the base stations are “at capacity” either because they are transmitting redundant data to recover Wmax 1 − pp 1 r − ≈ , . (4) from losses; or because they cannot effectively serve more g tcp min  1−  RTT pp RTT 5 + 2 pp than a few hundred active users. This may be costly as base 3 3 pp   q   stations are expensive to operate. One way to make sure that Note that unlike TCP/NC, TCP performance degrades propor- wireless networks are efficient is to ensure that, whenever base 1 tionally to p . stations are added, they are added to effectively increase the q Combining Equation (4) and rg−tcp ≤ rt(1−pp), we obtain goodput of the network. the values of rg−tcp for various rt, RTT , and pp as shown We studied the number of base stations nbs needed to in Figure 5b. As in Figure 5a, the values of rg−tcp are also improve the goodput rg to the users. It may seem that higher rg 5

15 15 15 15

10 10 10 10 bs bs bs bs n n n n 5 5 5 5

0 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 r (Mbps) r (Mbps) r (Mbps) r (Mbps) t t t t

(a) pp = 0% (b) pp = 1% (c) pp = 2% (d) pp = 5%

Fig. 6: The value of nbs from Equation (2) for TCP and TCP/NC with varying pp and p. Here, RTT = 100 ms, Wmax = 50, n = 1000, and µf = 3.2 MB. In (a), pp =0 and both TCP and TCP/NC behaves the same; thus, the curves overlap. Note that this result is the same as that of Figure 3a. In (b), the value of nbs with TCP for p =0.03 and 0.04 coincide (upper most red curve). In (c) and (d), the values of nbs with TCP for p> 0.01 overlap.

15 15 15 15

10 10 10 10 bs bs bs bs n n n n 5 5 5 5

0 0 0 0 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 r (Mbps) r (Mbps) r (Mbps) r (Mbps) t t t t

(a) pp = 0% (b) pp = 1% (c) pp = 2% (d) pp = 3%

Fig. 7: The value of nbs from Equation (2) for TCP and TCP/NC with varying pp and p. Here, RTT = 100 ms, Wmax = 50, n = 1000, and µf = 5.08 MB. In (a), the results for TCP and TCP/NC are the same. Note that this result is the same as that of Figure 3b. In (b) and (c), the value of nbs with TCP for p> 0.01 coincide (upper red curve). In (d), the values of nbs with TCP for any p all overlap. We do not show results for pp = 4% or 5% as they are similar to that of (d). necessarily increases nbs. Indeed, if there are enough demand REFERENCES (i.e. rg, p, or µf are high enough), we eventually need to [1] Cisco, “Cisco visual networking index: Global mobile data traffic increase nbs. However, we show that this relationship is not forecast,” 2011. [2] D. Kilper, G. Atkinson, S. Korotky, S. Goyal, P. Vetter, D. Suvakovic, necessarily true. When rg is low, each transaction takes more and O. Blume, “Power trends in communication networks,” IEEE time to complete and each user stays in the system longer. This Journal of Selected Topics in Quantum Electronics, vol. 17, no. 2, pp. degrades the user experience and delays the release of network 275 –284, 2011. resources dedicated to the user. This is particularly important [3] J. Padhye, V. Firoiu, D. Towsley, and J. Kurose, “Modeling TCP throughput: A simple model and its empirical validation,” in Proceedings as the number of active users each base station can support of the ACM SIGCOMM, 1998. is limited to the low hundreds. We observed that, given rt, [4] H. Balakrishnan, V. N. Padmanabhan, S. Seshan, and R. H. Katz, “A comparison of mechanisms for improving TCP performance over wire- achieving low rg may lead to a significant increase in nbs and less links,” IEEE/ACM Transactions on Networking, vol. 5, December an ineffective use of the network resources; while achieving 1997. high rg may lead to reduction in nbs. [5] M. Kim, M. M´edard, and J. Barros, “Modeling network coded TCP throughput: A simple model and its validation,” in Proceedings of We showed that, in lossy networks, the goodput rg observed ICST/ACM Valuetools, May 2011. may not closely match the amount of resources dedicated to [6] R. C´aceres and L. Iftode, “Improving the performance of reliable the user, e.g. rg ≪ rt. This is due to the poor performance transport protocols in mobile computing environments,” IEEE Journal of TCP in lossy networks. To combat these harmful effects, on Selected Areas in Communications, vol. 13, no. 5, June 1995. [7] Y. Tian, K. Xu, and N. Ansari, “TCP in wireless environments: Problems network providers dedicate significant amount of resources, and solutions,” IEEE Comm. Magazine, vol. 43, pp. 27–32, 2005. e.g. retransmissions and error corrections, to lower the loss [8] T. J. Hacker, B. D. Athey, and B. Noble, “The end-to-end performance rates. This, however, results in the base station transmitting effects of parallel TCP sockets on a lossy wire-area network,” in Proceedings of the IEEE IPDPS, 2002. at high throughput rt but little translating to goodput rg. We [9] A. Ford, C. Raiciu, M. Handley, S. Barre, and J. Iyengar, “Architectural showed that TCP/NC, which is more resilient to losses than guidelines for multipath tcp development,” IETF, Request for Comments, no. 6182, March 2011. TCP, may better translate rt to rg. Therefore, TCP/NC may [10] J. K. Sundararajan, D. Shah, M. M´edard, S. Jakubczak, M. Mitzen- lead to a better use of the available network resources and macher, and J. Barros, “Network coding meets tcp: Theory and imple- reduce the number of base stations nbs needed to support users mentation,” Proceedings of IEEE, vol. 99, pp. 490–512, March 2011. at a given rg.