MAC schemes - Fixed-assignment schemes

Transcription

1 MAC schees - Fixed-assignent schees M. Veeraraghavan, April 6, 04 Mediu Access Control (MAC) schees are echaniss for sharing a single link. MAC schees are essentially ultiplexing schees. For exaple, on an interface of a tie-space-tie circuit switch TDM ultiplexing is used, while FDM is used on interfaces of a WDM based optical circuit switch. The input stage of these switches have deultiplexers and the output stage has ultiplexers. Coparable are the TDMA and FDMA MAC schees used in cellular (wireless) networks. Siilarly, packet-by-packet ultiplexing is used on links connected to a connectionless packet switch. In such networks, there is no reservation phase and packets are siply sent by end hosts. Coparable are the rando-access MAC schees used on shared wireless or wired access links. Packets collide within packet switches, i.e., packets are lost when buffers run out of space, and recovery is through retransissions (e.g., TCP). Enhanced schees have the senders adjust their sending rates to lower the probability of packet loss. This is siilar to rando-access MAC schees where collisions occur and recovery is through retransissions. Adjusting the probability of retransissions is akin to adjusting the sending rate in TCP. TCP has stability probles as does Slotted Aloha. In this lesson we focus on TDM/FDM (equivalently, TDMA/FDMA).. M/M/ queue, M/M/ queue and M/M/ queues with -ties slower servers See Fig. below. In TDM/FDM sharing schees, the link bandwidth is divided aong (i) M/M/ queues (ii) M/M/ queue (iii) M/M/ queue / / µ/ µ/ µ/ µ/ µ / µ/ Figure :Copare different ways of sharing a link: the first two are typically used to odel TDM/FDM based sharing, while the third is typically used to odel packet-based ultiplexing - but the third odel could be used to odel space-division ultiplexing (another fixed-assignent schee) µ/ flows. In the first case, the arrival process is split. Exaple: typical grocery store queues where there is a separate queue for each checkout clerk. In the second case, the arrival process feeds one

2 queue, but there are long snaking queue. -ties slower servers. Exaple: airline check-in counters with the one µ T ( iii) (Exaple 3.9, page 70 [4]) () µ T () i (Exaple 3.9, page 70 [4]) () P Q, where and µ + µ ( ρ) P Q p0 ρ! ρ T ( ii) p 0 k 0 ( ρ) k ( ρ) k!! ( ρ) (see QT lecture) (3) How do these copare? Try the atlab progras. In coparing (ii) and (iii) in the QT lecture, we said under light loads (iii) is far better, and under heavy loads, they are both the sae (exaple 3.0 [4], page 77). Point to note here: in being obsessed with the high-load case, we ight have ignored the low-load case and wound up with the worse solution. How do these copare with the M/M// schee described in the ErlangB, Engset (Blocked Call Clearing - BCC) and Blocked Call Queueing (BCQ) lesson? If we can odel the TDM/FDM sharing on a link as an infinite-buffer queueing syste with servers, then the M/M/ odel (ii) in Fig. or the M/M/ odel (i) in Fig. could be used. With applications such as telephony, where the rate per tie-slot (in the case of TDM) or frequency (in the case of FDM) is fixed, the concept of dividing link capacity into servers akes sense. But with data traffic, if we are sending files using a TDM/FDM sharing schee, what should be? It could be, which eans the whole bandwidth is allocated to one file transfer; all others are queued. When a transfer copletes, the whole capacity is allocated to the next file transfer. Service tie is set to file transfer delays. In the ErlangB odel, the nuber of sources is infinite and there is no roo to queue calls. There are servers and the syste can have only up to jobs. The Engset forula is derived for the sae M/M// syste but with a finite nuber of sources. The BCQ odel could be viewed as an M/M//N syste, where N is the nuber of finite sources. In the BCQ odel, blocking probability is 0.

3 . Non-exponential service ties What happens if the service tie is non-exponential? Use M/G/ queues because the M/G/ queueing syste has not been solved analytically. M/G// has the sae solution as the M/M// syste (prove - see [0], section 5..). So what akes the use of M/G/ queue suitable for TDM/FDM odeling? Clearly, if odel (i) of Fig. is applicable, we can use This is how an FDM syste is odeled in exaple 3.6 on page 94 of [4]. M/G/ queues. FDM: There are streas of packets arriving at rate. Packets are constant length. Using FDM, where each strea gets of the link capacity, each packet needs tie units for transission. Each channel is an M/D/ queue. Since W ρ ( µ ( ρ) ) and ρ ( ) ( ) and µ, W FDM ( ) (4) If we assue that the service is per file (not per packet), we could use the M/BP/ odel (where BP is Bounded Pareto) for file size distributions. TDM: To analyze a TDM ode of link capacity sharing when the service tie is not exponential, we need a new odel: M/G/ queue with vacations [4]. What are vacations? If there are jobs in the queue, the server takes a vacation. Let V, V, be vacations taken by the server. These are iid r.v., also independent of interarrival ties and service ties. A new arrival will have to wait for the job in service to coplete or the ongoing vacation to coplete. Therefore W R ( ρ) is still valid (fro an M/G/ queue and PQ lecture), where R is the ean residual tie for copletion of the service or vacation in process when a custoer arrives. Follow derivation on pages 9-94 of [4] to obtain the ean waiting tie in this M/G/ queue with vacations as: W X V ( ρ) V (5) Apply this vacations odel to a TDM syste. Each frae has slots and slot/frae is dedicated to each strea (reeber that there are traffic streas). We keep the incoing traffic streas separate and have each feed a server of rate given we only have an M/G/ odel not an M/G/ odel. The ean waiting tie is the sae as in the FDM schee because the 3

4 arrival rate is and service rate is. The difference between FDM and TDM is as follows: in FDM, a whole carrier (operating at rate ) is dedicated to a single strea for all tie, while in TDM, all carriers are allocated for the duration of tie slot within a frae of tie slots). If a job (packet/file) arrives anytie during the -slot frae, it is as if the server is on vacation and it has to wait until the slot associated with the strea in question coe around. Vacation tie is units; therefore V and V, W TDM ( ). (6) ( ) Thus, ean waiting tie is ore in TDM than in FDM. However, total delay (waiting tie + service tie) presents a different picture: T FDM + (7) ( ) T TDM + T ( ) FDM --- (8) In TDM, the service tie is unit of tie. Even though waiting tie in TDM is greater, the service tie is a lot less (assuing with FDM. ), and hence the total delay with TDM is saller than Copare this analysis with the slide arked TDMA in ac.ppt. It states that data is sent at an accelerated pace when the assigned tie slot begins. In the above analysis, once the assigned tieslot coes around, service tie is only slot. 3. Overhead of aking reservations [4]: Deterine the axiu throughput S in packets/tie. Let be the axiu throughput in successful reservation packets per reservation slot. If slotted ALOHA is used for reservation, e. If TDM,. In TDM, each station is given one reservation slot in the reservation interval. Hence. Let v «be the size of reservation packets, when the size of data packets is. Thus > S - + v (9) 4

5 because v reservation slots are necessary for a successful reservation. Even with slotted ALOHA if v is very sall, S can be really close to. See ac.ppt for the rando-access schee used on the control channels to obtain the fixed-channel assignent for a call. Copare with the use of connectionless packet-switched networks in conjunction with circuit-switched networks to carry signaling essages (i.e., requests for bandwidth). Why is it better to have an STP connect DS0 based telephony switches rather than use associated signaling? 4. Appendix Consider slotted FDM, where transission can occur on each channel at only slot boundaries. This is odeled as an M/D/ queue with vacations. Slot duration is tie units (tie needed to eit one packet). If there are no packets in the queue for a given channel at the beginning of a slot tie, server takes a vacation in that slot. Thus V and V. If there is a packet, server serves it and there is no vacation. Therefore W SFDM W FDM (applying (5)). (0) T SFDM T FDM () References [] A. Leon Garcia and I. Widjaja, Counication Networks. McGraw Hill, 000. [] W. R. Stevens, TCP/IP Illustrated Vol., Addison-Wesley, 997. [3] M. Mathis, J. Mahdavi, S. Floyd, A. Roanow, TCP Selective Acknowledgent Options, IETF RFC 08, October 996. [4] D. Bertsekas and R. Gallager, Data Networks, Prentice Hall, Second Edition, 99. [5] K. S. Trivedi, Probability, Statistics with Reliability, Queueing and Coputer Science Applications, First Edition, Prentice Hall, ISBN r. [6] A. Leon Garcia and I. Widjaja, Counication Networks, McGraw Hill, 000, First Edition. [7] E. Pinsky, A. Conway and W. Liu, Blocking Forulae for the Engset Model, IEEE Transactions on Counications, vol. 4, no. 6, June 994, pp [8] R. Syski, Introduction to Congestion Theory in Telephone Systes, Oliver and Boyd, Edinburgh, 960. [9] Mischa Schwartz, Telecounications Networks, Protocols, Modeling and Analysis, Addison Wesley, 987. [0] D. Gross and C. M. Harris, Fundaentals of Queueing Theory, Wiley Series in Probability and Matheatical Statistics, 985. [] S. M. Ross, Stochastic Processes. [] Bob Boorstyn s notes. 5