An Analysis of TCP over Random Access Satellite Links

Transcription

1 An Analysis of over Random Access Satellite Links Chunmei Liu and Eytan Modiano Massachusetts Institute of Technology Cambridge, MA mayliu, Abstract This aer analyzes the erformance of over satellite links with random access. The system considered consists of large number of identical source-destination airs, each emloying at its transort layer and a random access scheme at its layer. Simle formulas that well cature the system erformance are given, and some imortant roerties of the system erformance are resented as well. Secifically, in order to analyze the system, we roose a simlified version of the system. We then develo formulas for obtaining the throughut of this simlified system as a function of various system and rotocol arameters. Based on these formulas, it is shown that the maximum ossible system throughut is /e, which can be achieved only when the system arameters satisfy a given condition. The otimal layer transmission robability at which the throughut is maximized is derived as well. Furthermore, the imact of varying system and rotocol arameters on the system erformance is analyzed. The results show that for systems with very small roagation delay or very large number of source-destination airs, a throughut of /e can be achieved by setting the layer transmission robability to its otimal value. However, when the number of users is (relatively) small or the roagation delay is (relatively) large, the maximum achievable throughut can be substantially smaller than /e. Although the analysis is based on the simlified system, simulations on the original system show that the formulas and the above results can be used to describe the erformance of the original system as well. I. INTRODUCTION In this aer we consider the erformance of over a shared satellite link with random access (i.e., Aloha). Satellite links are inherently different from terrestrial links, and the erformance of over satellite has received a great deal of attention over the ast decade. Most revious work has focused on the imact of the large roagation delays and high bit error rates associated with satellite links, [], [4], [5], [6]. Overviews are given in [3], [8]. This aer addresses the multi-access nature of satellite networks, and analyzes the erformance over satellite links when a ure random access scheme is emloyed at the layer. In articular, the system considered consists of large number of identical ersistent source-destination airs, each air emloying at its transort layer and each source emloying a random access scheme at its layer. In order to analyze the system, we roose an simlified system that differs from the above system in that the transort layers emloy a simlified version of window flow control. For this simlified system, an analytical model is develoed, and equations are derived for solving for the system throughut, idle robability and collision robability. Based on the equations, we show that the maximum achievable throughut is /e and give a necessary and sufficient condition for achieving it. The imact of system and rotocol arameters on the system erformance is obtained both analytically and numerically. Finally, the erformance of the original system is examined by simulations and the results are comared with those numerical results obtained from the formulas for the simlified system. The results show that although the analysis and formulas are based on the simlified system, they are a good aroximation of the original system and all the results obtained are valid for the original system as well. The aer is organized as follows: the next section gives a detail descrition on the systems and rotocols examined. In Section III we develo an analytical model for the simlified system. Using this analytical model, Section IV gives formulas for the system s throughut, the conditions for achieving the maximum throughut, as well as the otimal layer transmission robability. The imact of various system arameters on erformance is also examined in Section IV, both analytically and numerically, and simulation results for the original system are resented and comared with the numerical results as well. Section V discusses the imlications of the various assumtions and concludes the aer. II. SYSTEM DESCRIPTION The system we consider consists of N identical sourcedestination airs (SD airs), where N is large, as shown in Figure. Each of the N sources has unlimited number of ackets to be sent to its corresonding destination. All N sources share a common channel to the satellite. Each SD air emloys at its transort layer for congestion control urose, and all sources emloy a simlified version of ALOHA multi-access scheme at their layer for multiaccess urose. For simlicity, we ignore the timeout udate of based on its round tri time measurements, and assume that the timeout value is a random variable with mean TO slots. The ALOHA schemes will be described in detail later. For all SD airs, all ackets have the same length and each acket requires one time unit (called a slot) for transmission. The two-way roagation delay, defined to be the duration between the time when a acket is successfully transmitted by

2 the layer of the source and the time when its corresonding acknowledgement is received by the source, is a random variable with mean D slots. Here the randomness reresents the queuing delays and other random delays exerienced by the ackets. To focus on the imact of random access, assume that there are no other acket losses in the network excet losses due to layer collisions. Sat. TABLE I WINDOW EVOLUTION AND ROUNDS ACK window acket round first acket number of ackets size released in round in round 2 2, , , two-way roagation delay D S S2 SN D D2 DN Fig.. System with over Random Access The simlified version of ALOHA multi-access scheme works as follows: When one or more ackets are available in the layer buffer, the layer transmits the first acket with robability. If two or more ackets are transmitted in a given slot, a collision occurs and all ackets are assumed lost. If only one acket is transmitted in a given slot, the acket is successfully received. The difference between this scheme and ordinary ALOHA lies in that here we do not allow retransmissions of ackets involved in collisions at the layer. This assumtion is reasonable over a satellite link as retransmissions at the layer will almost certainly result in a time-out event, which can be easily verified. As with ALOHA, each layer of the system can be viewed as a queuing system, with all arrivals coming from its corresonding transort layer and geometrically distributed service time with mean /. In order to obtain an analytical model, we consider a simlified system that differs from the above system only in that it has a different window flow control (WFC) scheme. Its WFC scheme is similar to that in the congestion avoidance hase of but without the fast retransmit/recovery otion. In articular, it works as follows: Initially the window size W =. Uon an acknowledgement, the window size is udated to be W W + W. Uon a timeout, the window size is reset to be W. We also assume that the transort layer emloys a selective reeat retransmission rotocol and retransmits only those ackets that were lost. Furthermore, we assume that the timeout value is large enough that the robability of timeout due to queuing delay at the layer can be ignored. Recall that there are no losses other than collision losses. Hence, timeouts can only be triggered by layer collisions. The analysis henceforth will be based on the simlified system with these assumtions, and we therefore call the simlified system the system when there is no ambiguity. Nevertheless, the analytical results match well with the simulations for the original system with without timeout udate, as will be resented in Section IV. Including timeout udate into analysis is art of our future work. For convenience, we borrow the concet of round from [7] at the transort layer, with extension that includes the timeout signal: a round starts with transmission of W ackets, where W is the current window size of the congestion window. Once all ackets falling within the congestion window have been sent, no other ackets are sent until the first ACK for one of these W ackets or a timeout signal is received. This ACK recetion or timeout marks the end of the current round and the beginning of the next round. We further index the ackets sent and the rounds between two successive timeouts in order, i.e., acket k isthe kth acket and round k is the kth round. The ACK for acket k is called ACK k. Table I illustrates the window evolution and rounds between two successive timeouts for the simlified system. For examle in Table I, each time when the window size is increased by, there are two ackets released. Otherwise, there is one acket released. After the window size reaches 3, there are three ACKs received, namely ACK 3,4 and 5, before the window size reaches 4. Corresondingly, four ackets are released, namely acket 5, 6, 7 and 8. These are shown in column to column 3. Round 4 begins with ACK 4, which is the first ACK of ackets in round 3, and the first acket in round 4 is acket 7. ACK 7 thus becomes the first ACK of ackets in round 4 and marks the end of round 4 and beginning of round 5. This is shown in the 4th and 5th columns. By counting the number of ackets in round 4, we obtain 4, which is shown in the last column. III. PERFORMANCE ANALYSIS In the following we first consider the entire system including all SD airs, and then examine one SD air in isolation. The combined analysis yields formulas that together can be used to solve for the system erformance. A. Analysis at the System Level For every sender/receiver air, define the throughut as the number of ackets correctly received by the receiver er unit

3 time, and the send rate as the number of ackets sent by the sender er unit time. That is, the send rate includes both ackets that are included in throughut and the ackets that are lost due to collisions. By assuming equivalence between time average and ensemble average, the throughut is also the robability that in a slot, a acket is correctly received by the receiver. Similarly, the send rate is also the robability that in a slot, a acket is sent by the sender. In our system, on one hand, we have two layers, the transort layer and the layer. We therefore have definitions of throughut and send rate at both layers. On the other hand, the layers receive ackets only from their corresonding transort layers. Therefore, the send rate at both layers are the same. In addition, collision losses at the layers are the only losses in the system. So the throughut at both layers are the same as well. Henceforth, we use the name throughut and send rate to refer to the throughut and send rate at both layers. Besides, the throughut and send rate can refer to those of a single SD airs and those of all N SD airs. We call those for a single SD air individual throughut and send rate, and denote them by λ d and B d, resectively. We call those for all N SD airs system throughut and send rate, and denote them by λ s and B s, resectively. From the above definitions and discussions of throughut and send rate, we conclude that in our system, the system throughut λ s is the robability that there is exactly one layer sending a acket in a slot. For one articular SD air, the individual send rate B d is the robability that the layer of this SD air sends a acket in a slot. Since we have a large number of SD airs (N is large), we further assume that for a articular SD air, its state is indeendent of the state of other SD airs. We hence have the following relationshi between the system throughut, idle robability, collision robability and the individual send rate: λ s = NB d ( B d ) N NB d e NB d, P I =( B d ) N e NB d, P C = λ s P I. (a) (b) (c) where P I and P C are the idle robability and collision robability, resectively, and the aroximations hold when B d is small and N is large. On one hand, due to the indeendent assumtion, the above analysis is similar to that of a standard ALOHA system. The equations in () show that, as in an ALOHA system, the number of ackets correctly received in a slot can be aroximated by a Poison random variable, with the attemt rate NB d as its mean. The maximum ossible throughut can be achieved at NB d =, with the corresonding throughut, idle robability and collision robability being /e, /e and 2/e, resectively. On the other hand, different from an ALOHA system, NB d =is not always achievable in our system due to the transort layer window limitation, which will be analyzed in detail later, while in an ALOHA system, attemt rate can always be achieved with roer arameter settings. We henceforth call the system erformance at NB d = ALOHA erformance. Furthermore, under the indeendent assumtion, the effects of other SD airs on one articular SD air are aggregated into one arameter Q, defined to be the robability that all other N layers transmit no ackets in a slot. Since all SD airs are identical, we have: Q =( B d ) N e (N )B d e NB d. (2) Note that although the values of P I and Q are aroximately the same (Equation (b) and (2)), they have different hysical meaning. P I is for all N SD airs, while Q excludes the SD air considered and is for the other N SD airs only. Moreover, the indeendent assumtion also gives λ s = Nλ d and B s = NB d. B. Analysis at the session level As mentioned before, the effects of the other N SD airs on one articular SD air are aggregated into one arameter Q, the idle robability of all other N SD airs. This articular SD air can thus be modelled as a normal transort layer session with collision robability of each acket being Q. This section models one articular SD air as a renewal rocess and obtain an uer bound and a lower bound for its send rate, B d, in terms of Q. For one articular SD air, let s consider after a timeout, what haens before the first collision occurs. All ackets sent before the collided acket won t encounter a timeout, since they encounter no losses and the large timeout value assumtion ensures that their ACKs will be received before the retransmission timer exires. Whereas the collided acket will eventually encounter a timeout, since there is no mechanism other than the transort layer retransmission to recover this collision loss. Therefore, the first collision after a timeout causes a successive timeout. We introduce the concet of cycle to denote the interval between two successive timeouts. Moreover, since the transmissions of the ackets between the first timeout and the collided acket were successful, the transort layer sender will receive some ACKs after the collision and release some new ackets. The large timeout value assumtion ensures that the layer will finish the transmissions of these later released ackets before the second timeout signal. Thus uon each timeout signal, there is no acket in the buffer. In addition, according to the WFC scheme, the window evolves exactly the same after each timeout signal. The timeout signal sequence therefore forms a renewal rocess, and a cycle between two successive timeouts is an inter-arrival eriod of the renewal rocess. Let M be the number of ackets sent during a cycle and T be cycle length. Then by the renewal theory, B d = E[M] E[T ]. (3) To solve for B d, let s first consider E[T ]. Instead of deriving its exact exression, which requires comlicate queuing

4 analysis, we derive an uer bound and an lower bound. Let R be the number of successful round during a cycle and RT O be the timeout value that ends the cycle. Then T is the sum of the duration of these R rounds lus RT O. Each round takes at least the service time of its first acket, denoted by F k for round k, lus the two way roagation delay this acket exeriences, denoted by D k, for the sender to receive its ACK. We therefore have a lower bound for T, denoted by T L,tobe T T L = R k= (F k + D k )+RT O. Similarly, an uer bound for T, denoted by T U, can be obtained when ignoring the overlaing between the service time of each acket and D k of each round k. Mathematically, define Z to be the index of the acket that incurs the first collision and X k to be the service time of acket k. Then T T U = Z k= X k + R k= D k + RT O. Along with the lower bound T L,wehave R Z R (F k + D k )+RT O = T L T T U = X k + D k + RT O. k= k= k= By our multi-access scheme, the service time for each acket is geometrically distributed with mean / and indeendent of each other and R and Z as well. Thus by taking exectations of the above inequality, ( +D)E[R]+TO = E[T L ] E[T ] E[T U ]= E[Z] +DE[R]+TO. (4) Now let s consider E[M]. Recall that acket Z is the acket that incurs the first collision. This means that the revious Z ackets were successfully transmitted and have been or will be ACKed, while acket Z and thereafter won t be ACKed. According to the WFC scheme, new ackets can be released only uon recetion of ACKs. Therefore, M equals to, counting for the first acket, lus the number of ackets triggered by the Z ACKs. In addition in the WFC scheme, each ACK triggers the release of one acket, excet those ACKs that increase the window size by (i.e., the last ACK in a round) where two ackets are released (See Table I for illustration). Let I be the number of such ACKs in the Z ACKs, then M =+(Z ) + I = Z + I. Moreover, the window size is increased by er round. Therefore I = R, and E[M] =E[Z]+E[R]. (5) We now have bounds for E[T ] as in Inequality (4) and E[M] as in Equation (5) in terms of E[Z] and E[R]. Recall that Q is the robability that no other senders send their ackets in one slot and this event is indeendent of the state of the articular SD air. Therefore, each acket of the articular SD air incurs a collision with robability of Q and indeendent of each other. Z is thus geometrically distributed with Pr(Z = z) =Q z ( Q) and E[Z] =/( Q). By exloring the relationshi between R and Z, it can be shown that E[R] = k= Qk(k+)/2. For brevity, we omit the details. By combining them with Equation (3), Inequality (4) and Equation (5), we obtain the following bounds for the send rate B d : Q + k= Qk(k+)/2 ( Q) + D k= Qk(k+)/2 + TO = f L B d f U Q = + k= Qk(k+)/2 ( + D) k= Qk(k+)/2 + TO. (6) Notice that for large enough timeout value TO, both bounds are increasing functions of Q. Mathematically, this can be shown by taking the derivative of the bounds with resect to Q. Physically, from our derivation, the lower bound corresonds to the send rate of the following system: after sending the first acket of each round, the layer holds the transmission of other ackets until it receives the ACK of the first acket. This is how we obtain the uer bound for the duration T between two successive timeouts. Clearly, the send rate of this system increases with the idle robability of the other N SD airs Q. Similarly, the uer bound corresonds to the send rate of the following system: after sending the first acket of each round, the layer finishes the transmission of all other outstanding ackets before it receives the ACK of the first acket, that is, within time D k for round k. Thisis how we obtain the lower bound for the duration T between two successive timeouts. Again, the send rate of this system increases with the idle robability of the other N SD airs Q. To fully under the system behavior, the exectation of the collision window W z, defined to the window size when the collided acket Z is released (i.e., when the collision occurs), can also be shown to be: E[W Z ]=+ k=2 Again for brevity, we omit the details. Q k(k+) 2 2. (7) IV. SYSTEM PERFORMANCE Section III-A analyzes N SD airs together and gives one relationshi between the individual send rate B d and the idle robability of N SD airs Q, in Equation (2). Section III- B gives an uer bound and a lower bound of B d in terms of Q in Inequality (6). Based on these results, this section first gives bounds for B d and Q in terms of system and rotocol arameters, and then discusses the system erformance. A sufficient and necessary condition for the system to achieve the ALOHA erformance is also given, as well as the otimal layer transmission robability at which the throughut is maximized. The analysis is confirmed by the numerical results under different system and rotocol arameters. This section also gives the simulation results on the original system and comares them with the numerical results for the simlified system. Before roceeding, let s first introduce how we set the system and rotocol arameters in the numerical comutations

5 and simulations. For the systems, we consider the following data that is tyical for a satellite link: the two-way roagation delay of one transmission is around sec; acket size L is about 0000 bits; and transmission rate is around 00k - M bs. Converting these into the arameters in our system, we obtain that the two-way roagation delay is about 0-00 time slots. We therefore set the two-way roagation delay of ackets to be a uniformly distributed random variable with mean D being 0-00 slots and range of 4 slots. As mentioned before, the randomness reresents the queuing delay and other random delayed exerienced by ackets. The transort layer timeout value is also set to be a uniformly distributed random variable with mean TO. Similar to the timeout udate in, TO is set to be the average round tri time seen by the transort layer ackets, denoted by RT T, lus four times its standard deviation. Because of the large roagation delay of satellite links, when is not close to zero, we can ignore the queuing delays the ackets exeriences at the layer and aroximate RT T by the sum of acket service time at the layer and the twoway roagation delay. Recall that the acket service time at the layer is geometrically distributed with mean / and variance ( )/ 2. Therefore, we obtain TO + D +4. Notice that this setting of TO is used only in the numerical comutations and simulations. The following analysis of imact of arameters on the system erformance does not deend on this articular timeout value setting. It only requires that TO decreases with and increases with D, which should be true for other TO settings as well. Simulations are erformed on ns simulator for the original system. The timeout value is set in the same way described above. A. System Performance and Condition for ALOHA Performance Based on the Equation (2) and Inequality (6), this subsection shows how to obtain the system erformance given the system and rotocol arameters. and Q should be on curve g, as well as be inside the area between curve f L and f U. Therefore, B d and Q are on the section of curve g between curve f L and f U. The intersections of curve g with curve curve f L and f U thus give a lower bound and an uer bound for B d as well as a lower bound and an uer bound for Q. Denote them by Bd L and BU d, QL and Q U, resectively. Clearly, Bd L and BU d are solutions of the following two equations, resectively: B d = e NB d + k= e NB dk(k+)/2 + D ( e NB d ) k= e NB dk(k+)/2 + TO, (8a) e B d = + NB d k= e NB dk(k+)/2 ( + D) k= e NB dk(k+)/2 + TO. (8b) Figure 3 lots the system erformance, the system throughut, idle robability and collision robability, as a function of B d, given in Equation (). From the figure we can see that the bounds for B d actually give us the range of the system erformance. Secifically, if both bounds for B d, Bd L and BU d, lie within the same monotonic region of the throughut, i.e., within [0, /N ] or [/N, ] as in Figure 3, then their corresonding throughut, λ L s and λ U s in Figure 3, are also lower and uer bounds for the actual throughut λ s. Otherwise, Bd L [0, /N ] and BU d [/N, ], the actual throughut is close to the maximum ossible throughut /e. Furthermore, since P C and P I are monotonic with B d,the collision robability and idle robability corresonding to Bd L and Bd U are also bounds for P C and P I. As in Figure 3, PC L P C PC U and P I L P I PI U. We thus conclude that the two bounds given in Equations (8), together with Equation (), fully characterize the system erformance. Fig. 3. System Performance with B d Fig. 2. Bounds for B d and Q Figure 2 lots the relationshi between B d and Q as given in Equation (2), called curve g. Figure 2 also lots the bounds of B d, f L and f U, as given in Inequality (6). For clarity, the distances between the curves are exaggerated. The actual B d Our numerical results further show that in almost all cases, these two bounds are very close to each other. Table II gives some numerical results for different system and rotocol arameters N, D and. The difference between Bd L and Bd U is shown in the last column. Henceforth we aroximate the actual B d by Bd U and use B d and Bd U exchangeably. In conclusion, the system erformance is well aroximated by Equations (8b) and (). Now let us find a sufficient and necessary condition on which the ALOHA erformance, with the system throughut

6 TABLE II DIFFERENCES BETWEEN B U d AND BL d D N Bd U Bd L Bd U BL d /e, can be achieved. Recall that this erformance can be achieved if and only if NB d =. By lugging NB d = into Equation (8b) and noting that its solution for B d is unique (see Figure 2), we obtain the following sufficient and necessary condition for the ALOHA erformance: 2.0N 0.42D TO 0.42 =0. (9) Notice that TO is a function of D and. Also notice that due to the limit range of the actual arameters, the above condition cannot be always satisfied by adjusting one arameter with others fixed. For examle for very large D, the solution of the above condition for can be negative, while the actual transmission robability has to be nonnegative. In this case, the ALOHA erformance cannot be achieved by adjusting only. Overall, given the system and rotocol arameters, the system erformance can be solved from Equation () and (8b). Condition (9) is a sufficient and necessary condition for the system erformance to achieve the ALOHA erformance. Due to the limited range of system and rotocol arameters, this condition cannot be always satisfied by adjusting one arameter with others fixed. That is, the ALOHA erformance is not always achievable. The following subsections analyze how the system erformance changes with different system and rotocol arameters. B. Imact of Transmission Probability on System Performance First consider the imact of transmission robability on the system erformance. By the definition of f U (Inequality (6)) and noting that TO is a decreasing function of, f U increases with increasing. Thus as increases, curve f U in Figure 2 moves u. On the other hand, curve g is not a function of and remains the same. Therefore, the intersection of curve g and f U moves leftwards. Consequently, Bd U increases monotonically with. Recall that condition (9) is a sufficient and necessary condition for Bd U to achieve /N. Also note that [0, ] and when = 0, Bd U = 0. Therefore, when increases from 0 to, if the solution of condition (9) for, denoted by max, lies within [0, ], then Bd U increases from 0 to /N then to some number. Consequently, the throughut first increases, then decreases, with maximum /e (see Figure 3). Otherwise, Bd U increases from 0 to some number below /N. Consequently, the throughut increases monotonically with, and the maximum throughut is achieved at =. Similarly, since P I and P C is monotonic with B d, P I decreases monotonically with, and P C increases monotonically with. The above discussion actually gives us the transmission robability at which the throughut achieves its highest value, denoted by ot, as follows: ot = { max when max [0, ] (0) otherwise Moreover, if ot = max [0, ], then the system achieves ALOHA erformance. Otherwise, ot =, and the system erformance, with throughut below /e, can be solved from Equation (8b) and Equation (). Physically, when is very small, the send rate B d is very small (lies in [0,/N]. See Figure 3). Most times the system is idle and few ackets incur collisions. That is, the idle robability P I is high and the collision robability P C is low. Increasing increases B d and P C but decreases P I. Although this leads to more collisions, the number of idle slots also decreases, and the overall system throughut λ s increases. If with increasing, the send rate B d remains below /N after reaches, the throughut increases monotonically with (the case ot =). Otherwise, the send rate B d goes beyond /N after reaches a certain oint ( max < ), the system begins to incur too many collisions, and the throughut begins to dro. That is, in this case, the throughut first increases then decreases, with maximum /e achieved at max. Fig. 4. System Throughut as a Function of Transmission robability By solving Equation (8b) and (), Figure 4 shows the numerical results for the throughut λ s as a function of when D =0and N =20as well as when D =50and N =20. The first case (D =0) corresonds to the case ot <, and as exected, the throughut first increases with then decreases. The second case (D =50) corresonds to the case ot =, and the throughut increases monotonically with. For comarison, the throughut corresonding to Bd L is also shown. We can see that the two throughut curves corresonding to Bd U and BL d are very close to each other, which further confirms that using the throughut corresonding to Bd U to aroximate the actual throughut is good. Figure 4 also lots the simulation results for the original system with the same sets of arameters. It can be seen that the numerical results based on Equation (8b) and () match well with the simulation results for the original system.

7 Moreover, both the numerical results and simulation results for the above cases indicate that the exected collision window decreases monotonically with, with value below 2 for 0.. For brevity, the curves are not shown here. C. Imact of Two-way Proagation Delay on System Performance Analog to the above analysis for, it can be shown that Bd U decreases monotonically with D. For brevity, we omit the details. Furthermore, when D ranges from to infinity, if the solution of condition (9) for D, denoted by D max, lies within [, ], then the ALOHA erformance is achieved at D max. In this case both larger D and smaller D lead to a throughout below /e. Otherwise, D max / [, ], and the throughut increases monotonically with decreasing D and is always below /e. Physically, a low throughut can be either due to too many idle slots or too many collisions. Consequently, for normal and N with D max [, ] and starting from D max, increasing D leads to too many idle slots and reducing D leads to too many collisions. Therefore both result in monotonic decreasing of the throughut with D. While for very small and N with D max / [, ], there are always too many idle slots no matter how small D is. Reducing D thus reduces the number of idle slots and always increases the throughut. Fig. 5. System Throughut as a Function of Two-way Proagation Delay Figure 5 shows the numerical results for the throughut λ s as a function of D [0, 00] when N =20and =0.7 as well as when N =0and =0.7. In the first case (N =20) the ALOHA erformance in achieved around D max 25, while in the second case (N =0), even when D =0, there are still too many idle slots and the throughut is still below /e. To illustrate the imact of adjusting rotocol arameters on the system erformance, for each D in each case, max and ot are also calculated from condition (9) and Equation (0) and the resulting throughut is lotted in Figure 5 as well. The figure shows that for large D, max / [0, ] and ot =. The resulting otimal throughut is below /e. While when D is relatively small, the highest ossible throughut /e can always be achieved by setting the transmission robability = ot = max. That is, we can always lower the transmission robability to counterbalance the increasing collisions resulting from smaller D. Fig. 6. Comarison between Numerical and Simulation Results - Throughut vs Two-way Proagation Delay Simulation results for the original system with the same sets of arameters are lotted in Figure 6. For comarison urose, the numerical results obtained from Equation (8b) and () and lotted in Figure 5 are relotted here as well. Again, the figure shows good match between them. Moreover, the numerical and simulation results for all cases also show that the exected collision window is a nondecreasing function of D and is always below 3. Actually even when D is further increased to 5000, the exected collision window is still below 8. Again for brevity, the curves are not shown here. D. Imact of Number of Users on System Performance The analysis for the imact of the number of users N on the system erformance is also analog to that for the transmission robability. Differently, when N increases from to infinity, curve f U remains the same while curve g moves downwards. Consequently, the intersection of the two curves moves leftwards, and Q L decreases. Recall that NB d increases as Q decreases (Equation (2)). Therefore, NB d increases monotonically with N. Moreover, it can be easily verified that, for any [0, ], D and TO + D (the actual TO should be at least the average RTT of ackets, which is at least + D), the solution of condition (9) for N, denoted by N max, is always greater than. Therefore, as N ranges from to infinity, NB d increases from some value below to infinity. Consequently, the system throughut first increases then decreases, with maximum close to /e achieved at the integer closest to N max. Physically, larger N makes the ackets in the system more dense, which means less idle slots and more collisions. When N is close to N max, the idle slots and collisions reach a balance and the throughut is close to its highest value /e. Further decreasing N or increasing N results in either too many idle slots, or too many collisions, both of which lower the throughut.

8 Fig. 7. System Throughut as a Function of the Number of SD Pairs Figure 7 shows the numerical results for the throughut λ s as a function of N [0, 40] when D =30and =0.7 as well as when D =30and = ot. As mentioned before, the mean timeout value is set to be TO = + D +4.The throughut of the curve with =0.7has maximum value of /e, and first increases then decreases, which confirms the above analysis. The curve with = ot further shows that for large enough N, the maximum ossible throughut /e can be achieved by adjusting the transmission robability according to condition (9). That is, in order to achieve higher throughut, we can always lower the transmission robability to counterbalance the increasing collisions resulting from larger N. Simulations for the original system with the same sets of arameters are also erformed and the results are comared with the numerical results from Equation (8b) and () as well. Again, they have a good match. Besides, both of them give the exected collision window size in all cases being below 3. Again for brevity, the curves are not shown here. In summary, since the numerical curves obtained from the equations match well with the simulation results for the original system, we conclude that the equations can be used to describe the erformance of the original system. In articular, given the system and rotocol arameters,, D, N and TO, Equation () and (8b) give us the system erformance. Condition (9) is a sufficient and necessary condition on the arameters for the system to achieve ALOHA erformance, which has maximum ossible throughut /e. The otimal transmission robability at which the throughut can achieve its highest value is given in Equation (0). For systems with very small D and/or very large N, the ALOHA erformance can always be achieved by setting to its otimal value. For fixed, a system with very large D and/or very small N has a smaller throughut than a system with relatively smaller D and/or larger N due to too many idle slots. On the contrary, a system with very large N has a smaller throughut than a system with relatively smaller N due to too many collisions. Furthermore, in all cases in our numerical comutations, the exected collision window is below 4. This is because of too many collisions due to random access. V. DISCUSSIONS Our urose is to analyze the erformance over satellite links with random access. We focus on its window flow control mechanism and deliberately disregards other asects of, such as RTT measurements and estimation and timer granularity. The system analyzed has a window flow control scheme that differs from that in in two asects: first, we only considered the window evolution in the congestion avoidance hase; and second, we ignored the effect of dulicate ACKs for fast retransmission/recovery. Our simulations show that even with these differences, the analysis gives good rediction on the erformance of the original system. Here we discuss the effects of these two differences. Due to the large number of collisions that result from the random access rotocol, the system original with has a very small congestion window size. This is confirmed by the simulations, which shows that it is normally below 4 and below 8 for extremely large D. As a result, the window threshold will also be very small (below four). With such a small window threshold, the window evolution with only congestion avoidance hase is close to that with both slow start and congestion avoidance hase. Furthermore, the authors in [2] show that for a small congestion window such as eight, fast retransmission/recovery can seldom be entered into if there are multile losses within the window, which is the case in our system due to collisions. Therefore the window flow control scheme analyzed is a good aroximation of that in in our system as well. For the random access scheme, we did not consider collision recovery at the layer. Over a large delay satellite link, this assumtion is also reasonable as any attemt at delayed retransmission will result in a time-out event with very high robability. Hence, it is sensible to leave such retransmissions to higher layers. In addition, we did not take into account the timeout udate in. One future direction is therefore to consider collision recovery at the layer as well as to model the timeout udate at the transort layer. REFERENCES [] H. Balakrishnan, V.N. Padmanabhan, S.Seshan and R.H. Katz, A comarison of mechanisms for imroving erformance over wireless links, IEEE/ACM Transactions on Networking, Vol. 5, , December 997. [2] K. Fall and S. Floyd, Simulation-based comarisons of Tahoe, Reno, and SACK, Comuter Communication Review, 996. [3] N. Ghani and S. 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