Initial exploration into the benefits of channel diversity for finite user scenarios.

1. Initial exploration into the benefits of channel diversity for finite user scenarios. Soumya Sen Advisor: Prof. Roch Guerin MNLab, UPenn

2. Section I Work Overview

3. Introduction • Channel performance may be affected by phenomenon that are: • Extrinsic: collision from users transmitting at the same time • Intrinsic: channel condition. Extrinsic phenomena can be mitigated by scheduling, static channelization. Our interest is in simple random access protocol, and using efficient user group sizes (dictates the amount of diversity) to maximize performance in various scenarios.

4. Motivation • How to group users & available frequencies? • Is it better to form ‘smaller’ or ‘larger’ groups? How does this decision depend on load, user population, available resources, message length (in packets)? • How does the condition of the channels (burst length) affect these groupings? Does coding help in improving performance in such cases? [under investigation]. • How are these results affected if different medium access protocols are used? [possible future direction]

5. It is a well known Random access protocol that has been used and studied widely for Satellite networks [1]. Slotted Aloha has also been considered for Mobile radio [2]. Slotted Aloha is a ‘possible direction to explore’ in UWSN due to high propagation delay scenario in UWSN, quite similar to that of Satellite networks. [3] Fiber optic networks have also used slotted Aloha and coding in scenarios with propagation delays larger than in satellite channels. This motivates its investigation for implementation in UWSN. [4,5] But in UWSN, performance of Slotted Aloha exhibits same utilization as non-slotted Aloha in UWSN.[6] M. Dippold, “Performance of a slotted ALOHA access scheme for packet transmission on the landmobile satellite channel,” Proc. 8th Int. Conf. on Digital Satellite Commun. M. Zorzi, “Mobile radio slotted ALOHA with capture and diversity”,INFOCOM '95. Jun-Hong Cui, “Challenges: Building Scalable Mobile Underwater Wireless Sensor Networks for Aquatic Applications”. R.Murali and B.L.Hughes. “Random Access with Large Propagation Delay”. IEEE/ACM Trans. Networking, 5(6):924–935, Dec. 1997. Jim Partan, Jim Kurose et al,”A Survey of Practical Issues in Underwater Networks”. Luiz Filipe M. Vieira et al, “Analysis of Aloha Protocols for Underwater Acoustic Sensor Networks”. Is considering Slotted Aloha useful?

6. Slotted Aloha protocol features • Random-access protocol • Send out packet at the beginning of a slot whenever data is available • A collision may occur from any simultaneous transmission from users in the same slot and on the same frequency. • Packets received in error are retransmitted. • Backoff algorithm for slotted aloha (Kleinrock-Lam 1973) – retransmit a collided packet randomly into one of k future time slots. (Poisson process assumption implies k→∞).

7. Slotted Aloha throughput: discussion of existing results For Slotted Aloha: S=G{1-(G/M)}M-1[Finite user case](Reference: Queuing Systems, Kleinrock) S=Ge-G [Infinite users](Roberts), S=Ge-2G for unslotted (Abramson). where S=Throughput, G=average channel traffic i.e. load - ARPANET satellite system (1972) In our model of Slotted Aloha, for k=1, we had throughput, Psuccess=p{1-(p/F)}M-1 [Finite user model] where F= num. of freq, M= num. of users, p= probability that the user is active in any given slot. Eff. Throughput, S = (kp/[1-p(1-k)])*(Psuccess/p) [k=1 here] Max. throughput at load=1. (ref graph, max throughput= 36.8%)

8. Additional considerations in our model • An user, whenever active, may choose to transmit on any of the available frequencies (randomly) in the beginning of every slot. • All users are identical and independent. • A message consists of multiple packets (k packets in our case) and this message length (k) is same for all users. • Whenever an user becomes active it transmits all the ‘k’ packets of the message over the next ‘k’ consecutive slots. • In absence of coding, a message is said to be correctly delivered if all the ‘k’ packets are transmitted without collision from transmissions of other active users. • Channels are in good condition always (in the models we will discuss today, extensions are on way to have GE model for channels). • Retransmissions are not considered.

9. Our model parameters • M= total number of users placed in each user group • F= total number of frequencies alloted to each user group. • k= number of packets per message in no coding scenario. • p= probability that an user becomes active at the beginning of any slot and continues to transmit for the next ‘k’ slots.

10. Section II Analytical Formulations

11. Starting with the simplest case: Messages have only one packet, k=1 In our model of Slotted Aloha, for k=1 i.e. each message is only one packet in size, we had throughput: Psuccess=p{1-(p/F)}M-1[Finite user model] where F= num. of freq, M= num. of users, p= probability that the user is active in any given slot. Even more simple case if: F=1 (i.e. M users share 1 frequency) Psuccess=p{1- p}M-1 Let us delve deeper now… k >1 i.e. each message has ‘k’ packets in it and it takes ‘k’ consecutive slots to transmit.

12. Markov chain realization of the model

13. Markov model when multiple frequencies are present • Each state (except idle state) now splits up into ‘F’ different states, one for each frequency. • States for this chain: the user is transmitting ith packet of the message on the jth frequency.

14. Analyzing a simple scenario: single frequency case (but k >1)

15. Analyzing the multiple frequency case and message with multiple packets (K>1)

16. Possible configuration of another user who was in state ‘0’ at the beginning of transmission from the node under consideration as time progresses

17. Possible configuration of another user who was also in state ‘1’ at the beginning of transmission from the node under consideration

18. Possible configuration of another user if he was in state ‘2’ at the beginning of transmission from the node under consideration

19. Possible configuration of another user if he was in state ‘3’ at the beginning of transmission from the node under consideration

20. Identifying the patterns and developing the formula

21. Obtaining a more compact form:

22. Some sanity checks • Substituting F=1: • Substituting k=1: • Substituting k=1, F=1:

23. Defining Throughput & load • On an average the expected number of slots for which the user is inactive is given by: • So for every active ‘k’ slots the user transmits, there is an inactive period of (1-p)/p slots on average. • So if the conditional probability, Psucc= Psuccess/π1 Load on the system is given by:

24. Section III Results: Plots & Simulations

25. Analyzing efficient user-group formation for different scenarios Scenario: Each message takes only one slot to get transmitted i.e. k=1, no coding is used, channels are always in good condition. Only collisions from simultaneous transmission affect throughput. Access protocol is slotted Aloha.

26. Proposition 1: (k=1) Smaller groups are better for ‘low’ values of p.

27. Analytical Treatment Proposition 1:Forming smaller sized groups are better for “low” values of ‘p’ i.e. p→0. Proof: For a fixed value of p and a fixed ratio r=M/F , we have throughput given by: Let F2>F1, and M1=rF1 & M2=rF2, we have: So y2<y1 for small p. So it is better to form smaller sized group {M=r, F=1} for k=1 at low values of p.

28. Proposition 2: (k=1) At high load values, it is better to form larger groups if M/F ≥ 2, else smaller groups are preferable.Case 1: M/F < 2

29. Proposition 2: (k=1) At high load values, it is better to form larger groups if M/F ≥ 2, else smaller groups are preferable.Case 2: M/F ≥ 2

30. Analytical Treatment Proposition 2:For a fixed ratio, r=M/F, and k=1, it is better to form large groups for high values of p if r ≥2, else it is better to form smaller groups when 1≤r<2. [if the continuous function is increasing monotonically, then the sampled/discrete point values are also increasing in nature]

31. We observe that is an increasing function that saturates to at a value of 2 for F→∞. • Correspondingly, y’ is negative for and in this case it is thus better to form smaller groups at p=1. • Analysis of behavior of to prove that it saturates to 2 for F→∞. • Analysis for r <2: For integer values of M and F, the largest r value that can be less than 2 is given by: (2F-1)/F.

32. Proof & analysis (contd..) Result: M/F ratio value of 2 is a critical value that determines suitability of various group size at high load values. Other obvious results: As M/F ratio value increases, throughput decreases.

33. Scenario: Each message takes multiple slots to get transmitted i.e. k>1, no coding is used, channels are always in good condition. Only collisions from simultaneous transmission affect throughput. Message is said to have been successfully transmitted if and only if all the ‘k’ packets reached without collision. Access protocol is slotted Aloha.

34. Observation of behavior of M/F ratio around value of 2 for k>1Case 1: r <2

35. Case 2: r ≥2and observation of effect of M/F ratio for K=2

36. Effect of increasing the value of k

37. Comments based on observation: • It is still better to form smaller groups for low values of p even for k>1. • The value of r=M/F of 2 still remains the an interesting value that determines whether smaller or larger groups may be preferred at high loads. • Increase in M/F ratio decreases throughput. • Increase in k value decreases throughput. • For values of k>2 or for k=2 and r>3, the throughputs are so highly affected by the increasing load that it turns out that smaller groups are always preferred under such scenarios.

38. Concluding remarks for the previous scenarios Smaller user groups are preferred in many occasions which implies that though grouping into larger groups bring in possible benefits of larger channel diversity, such gains are lost to the adverse effect of increased number of users that now compete for those resources and collide with each other’s transmissions (extrinsic channel disturbances), thereby bringing down the effective throughput of the system.

39. Simulation verification of the graphs

40. Section IV Present focus & directions

41. New questions.. • What happens if we add coding on top of these scenarios? • Becomes analytically intractable. • Exponential growth in the number of states and combinatorial calculations. • Simulation based approach shows that when channels are in good condition then addition of coding increases load (overhead increase) and all the benefits of diversity are lost to it. So will it be beneficial to consider coding if channels were suffering from intrinsic disturbances? If so, what size of user groups will perform well in those scenarios?

42. Other approaches for present and future extensions: • Analyzing the behavior of coding based scenario i.e. addition of coding always deteriorates performance by trying to model equivalent M/D/1 or M/M/1/∞ systems and observing their behavior. • Exploration of intrinsic channel quality and interplay of channel diversity benefits and coding over these channels (modeled as Gilbert-Elliot 2-state model) • Extending the work to other protocols as well like CSMA.

43. Thank you!