# Packet Queueing Delay in Wireless Networks with Multiple Base Stations and Cellular Frequency Reuse

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3 This section derives packet queuing delays in the base station. Since the probability of successful reception Q(r) decreases with increasing propagation distance r, the number of (re-) transmission attempts, M, statistically increases with increasing r. The service time S per packet is S = M T L with T L the duration of a time slot. The bit rate per Hz is denoted as η r, and the total system bandwidth is B N. So, T L = L / (η r B T ) = L C / (η r B N ) where B T denotes the bandwidth per cell with B T = B N /C. The maximum effective throughput in user bits per second per base station per Hz, is (1) The probability of successful reception of a data packet depends on the location of the terminal and on the activity of interfering base stations. For discontinuous transmission, the probability that an interfering base station is active is equal to the probability that its queue is non-empty. In a spatially uniform system, this probability is identical for all base stations. The number of packets waiting in the base station is modelled as an M/G/1/ queue. Service times conditional on r are geometric with mean T L /Q(r) but the unconditional distribution of service times is not a closed-form expression. We approximate the service time of successive packets as i.i.d. random variables. This assumption, however, ignores the fact that the service times of successive packets may be correlated. For instance if one packet experiences heavy interference from transmissions by anther base station, it is more likely that the next packet will also see an active co-channel base station. Furthermore, two adjacent base stations may continue to attempt to transmit to terminals at unfavorable locations. We neglect this effect in the following analysis as a more detailed investigation would require us to compute delays in an infinite field of mutually interacting queues. This problem is notoriously hard to solve. But in section 7 we will propose an access scheme that can resolve these potential instability problems. Moreover, appendix A evaluates the effect of correlated retransmissions. The mean value of the service time, expressed in number of slots and averaged over all locations in the cell, is (2) The second moment is (3) The expected queuing delay is found from the Pollacek-Khintchine expression for M/G/1/ queues, namely 3

4 (4) 3 Results A relevant performance measure of wireless data networks is the expected number of required transmission attempts versus terminal location. We used a typical modulation technique with receiver threshold z = 4 (6 db), UHF groundwave propagation with β = 4 and shadowing with a spread of 6 db (σ s = 1.36). Service times in seconds can be obtained by multiplying resultsthis by L / (η r B N ). Table 2 Maximum throughput S per base station for various cluster sizes C. C λ/η r Service Time CE[M] CE[M r=0.91] Results on the queuing delay in the base station, e.g., as a function of the message arrival rate λ/η r per cell, shows that C = 1 is optimum. The relatively high bandwidth in each cell allows the base stations to empty their queues fairly rapidly, which reduced interference to other cells. In contrast to the assumptions in the above analysis, shadow attenuation can be highly correlated for small antenna displacements. In such case shadow attenuation is likely to be almost identical during all retransmission attempts. Moreover, at high traffic loads the busy periods of each cochannel base station are likely to be much longer than the service time experienced by an individual packet. For the limiting case of identical shadow attenuation and identical interference situations during all retransmission attempts, E[ M] is derived in Appendix A. Table 3 shows that under heavy traffic, C = 1 does not give favorable performance. 4

6 4 Concluding Remarks The analysis reported here revealed that efficient mobile packet data transmission requires entirely different spectrum reuse than telephone nets. To optimize spectrum efficiency, user capacity and network performance, presumably a new class of access schemes is needed that dynamically combine random access within one cell with protection against interfering signals from other cells. Dynamic Channel Allocation (DCA) is known to provide a means to share bandwidth-timespace resources in a more dynamic and efficient way. However, DCA primarily works on a session by session basis, whereas bursty teletraffic is presumably best supported through access schemes that assign radio resources on a packet by packet basis, or at least on a burst by burst basis. Under certain assumptions, Contiguous Frequency Assignment (CFA), i.e., cluster size C = 1, can support approximately 0.4 bit/s/hz/cell. This appeared substantially more efficient than cellular frequency reuse with C = 3, 4, 7,.... CFA also provides the smallest packet delay at a given spatial packet throughput intensity. This suggests that, in order to ensure minimum delay at maximum user capacity, mobile radio data networks should be designed with much denser frequency reuse than typically used to ensure an outage probability on the order of 10 2 or This, however, results in low signal-to-interference ratios and large packet loss probabilities, which have to be addressed by appropriate data-link protocols. In extreme cases, that transmissions to a particular terminal always see the same interfering base stations, so the number of required transmission attempts may become prohibitively large for certain terminals. This could have a detrimental effect on the entire network performance and it has motivated us to start developing new 'spatial' random access protocols. REFERENCES [1] L. Kleinrock, "Queuing systems. Vol. 1: Theory; Vol 2: Computer applications", New York: Wiley, 1975, [2] D. Bertsekas and R. Gallager, "Data Networks", Prentice-Hall International, Inc., London, [3] A.S. Tanenbaum, "Computer Networks", Prentice-Hall International Editions, London, [4] J. Walrand, "Communication Networks: a first course", Aksen Associates, Series in Electrical and Computer Engineering, Homewood, IL, [5] J.P.M.G. Linnartz, "Narrowband landmobile radio networks", Artech House, Norwood, MA,

7 Appendix A: Number of transmission attempts with slow shadowing In contrast to the assumptions in the main body, we assume identical interference situations during all retransmission attempts of a particular packet. The number of interfering co-channel base stations I is treated as a binomial random variable ( I = 0, 1,, 6) with mean 6 P ON, where I is constant during the service time of an individual packet. Given the local-mean powers of the wanted and all active interfering signals, the expected number of retransmissions can be expressed as (5) If the local-mean powers of signals from interfering base station are i.i.d. log-normally distributed, we find, after solving the I-fold integration, (6) Rewriting the product into a sum of I-terms and using the property that in a log-normal channel (7) we find the closed-form expression (8) which we average over the binomial probability mass function of I. If the position of the receiving mobile is uniform within the cell, the expected number of (re-) transmissions per message is (9) 7

8 8

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