Maximum Number of Users Which Can Be Served by TETRA Systems

Maximum Number of Users Which Can Be Served by TETRA Systems Martin Steppler Aachen University of Technology Communication Networks Prof. Dr.-Ing. Bernhard Walke Kopernikusstr. 16 D-52074 Aachen Germany Phone: +49-2 41-80 58 28 Fax: +49-2 41-88 88-2 42 E-mail: martin@steppler.de WWW: http://steppler.de ØØ This paper provides results regarding the maximum number of users which can be served by TETRA systems depending on both, random and reserved access. Furthermore, the two different methods of V+D random access are discerned, the rolling access frame and the discrete access frame. The results are based on two different Markovian models which are also presented. In case of one MCCH, one access code and a typical traffic load, up to Æ = 556 mobile stations can be served by one base station at the same time consuming Ò = 6 frequencies with Å = 23 servers, i.e. time slots. I. INTRODUCTION A nationwide Terrestrial Trunked Radio (TETRA) system based on the protocol stack Voice + Data (V+D) [1, 2] is about to be implemented in Germany and other European countries. When planning a cellular network the number of base stations (BS) and the cell sizes are directly dependent on the maximum number of active mobile stations (MS) which can be served by one BS. In Sec. II formulae are derived in regard to the mean throughput, the mean backlog and the mean packet delay of a Slotted-ALOHA system with a finite number of active users. These formulae are based on a Markovian model. Backlogged packets have to be repeated with a given rate. Sec. III defines the upper and lower boundary of the repeat rate depending on V+D random access parameters and the choosen V+D random access method, i.e. rolling access frame (RF) or discrete access frame (DF). Then, the maximum number of MSs regarding random access is presented for a given traffic load mix in Sec. IV. After this, the second Markovian model is introduced to the reader in Sec. V, which is used to determine the waiting probability and the waiting time of MSs after successful random access. The typical mean transmission time of voice and data packets is then estimated in Sec. VI. Finally, the maximum number of MSs is presented which can be granted reserved access at the same time. Then, the maximum numbers of random and reserved access are compared to yield the final result given in Sec. VIII. II. ESTIMATION OF PERFORMANCE PARAMETERS OF S-ALOHA SYSTEMS WITH FINITE POPULATION Let Æ be the number of identical MSs generating packets per time slot (TS) according to a Poisson distribution with mean interarrival rate [ 1 /TS]. New packets are transmitted immediately. Repeated trials are not contained in. Thus, is the send rate, owing to which one MS generates a new packet after having successfully transmitted the previously generated packet. In case of stationary equilibrium, the total send rate of all MSs equals to Æ. Furthermore, assume that collided packets are buffered in the resp. MS and are repeated with rate [ 1 /TS] at all subsequent random access opportunities. This is why the waiting time Û is geometrically distributed and its mean value yields to Û Û ½ ½ ½ µ ½ ½ ½ ÌË (1) The transmission duration (one TS) of a collided packet is contained in the above waiting time. The state of an S-ALOHA system can completely be described by the current number of collided MSs. previously collided packets are buffered in MSs in state. Assume that these MSs are blocked and do not generate new packets. Each blocked MS decides on its own after an random waiting time, when to resend the buffered packet. The packet is sent with probability. The current TS is left out with probability ½. Besides this traffic load resulting from packets per TS, Æ non-blocked MSs generate new packets with the mean send rate Æ µ. The random variable (RV) number of buffered packets in the system is discrete and changes each TS, because new packets can collide and previously collided packets can be successfuly transmitted. In each state the RV may increase in the range ½ Æ, because several packets may collide in each TS, see Fig. 1. p p p p p p 0,n 2,n 0,3 n 2,n 0,n 1 0,n p 1,n p 0,2 p p p p 1,2 2,3 n 2,n 1 n 1,n 0 1 2 N 1 N p p p p p 1,0 2,1 3,2 n 1,n 2 n,n 1 p 0,0 p 1,1 p 2,2 p n 1,n 1 p n,n Figure 1: State diagram of an S-ALOHA system with finite population Due to the fact that only one collided packet can be transmitted successfully per TS, there are only back transitions to the previous state in the above Markov chain. There is no transition from state 0 to state 1, because newly created packets have to collide with at least one further packet and thus the next state is 2. Let be the number of newly generated packets of Æ non-blocked MSs and the number

of repeated trials of blocked MSs, then the transitions probabilities Ô (from state to state ) of the above Markov chain are: Ô ¼ for ¾ È ¼ È ½ for ½ ½ Æ ¼ Ô for È ½ È ½ for ½ È for ¾ with ¾ Æ ¼ ¼ Æ and and Ô µ È ¼ ½ µ Æ (2) È ½ Æ µ ½ µ Æ ½ (3) Æ È ½ µ Æ È ½ ½ µ ½ È ¼ ½ µ (4) È ½ ½ ½ µ Æ ¼ Ô Æ ¼ Ô µ Ô and Æ ¼ Ô µ ½ We can get the mean throughput Ë by weighting the throughput of each state Ë µ, i.e. (5) Ë µ È ½ È ¼ È ¼ È ½ (6) with the resp. state probability Ô µ. The mean number of blocked MSs, i.e. the backlog, and with LITTLE s result [3] the mean packet delay are calculated accordingly: Ë Æ ½ Ë µ Ô µ Æ ½ Ô µ III. ESTIMATION OF THE REPEAT RATE Ë (7) The V+D random access parameter waiting time (WT) is defined to be the number of a BS s answering opportunities on the downlink since an MS s random access which must elapse before an MS may consider its random access to have failed and may repeat it. For this reason, the MS then chooses uniformly distributed one out of à TSs of the next RF. Thus, the mean waiting time yields to Û Û Ï Ì Ã¾ [TS] (8) Assuming that the throughput Ë and the packet delay are only sensitive in regard to their mean values and not to their complete distribution functions [3, 4], Eqs. 1 and 8 can be set equal: ½ Ï Ì Ã¾ (9) Within the parameters of ½ TS Ï Ì ½ TS and ½ TS à ½¾ TS [1] the range of the RF repeat rate is ½ ½ ÌË ¾ ½ ÌË (10) If a BS uses the DF access method, then additional ¾ Æ ¼ TSs may elapse between two non-overlapping access frames. Thus, the mean waiting time is Û Û Ï Ì Ã [TS] (11) The DF repeat rate is with Eqs. 1 and 11: ½ Ï Ì Ã (12) This is why the DF repeat rate lies in the range ÐÑ ½ ½ ½ ¼ ½ ÌË ½¾ ½ ÌË (13) IV. MAXIMUM NUMBER OF USERS IN REGARD TO RANDOM ACCESS In [5] ten different scenarios are defined for the comparison of TETRA systems. The airport scenario is the one with the highest traffic load. The covered area is ¼ km ¾ and the expected MS density is ¼ ½ Ñ ¾. The interarrival rate of speech connection requests per MS is considered to be ½. Furthermore, MSs are expected to send out 100 byte short data messages at rate ¾¼ ½ and 2 Kbyte middle data messages at rate Ñ ¼ ½. The total sum Ø of all these arrival rates is as follows: Ø Ñ ¾½ ½ (14) The above arrival rate of new packets is used throughout the rest of this paper. According to [1], each TS of the Main Control Channel (MCCH), i.e. the first TS of a 4 TS frame, provides two random access opportunities (RAO). If a BS only uses one access code (AC), then there are two RAOs per frame. This is why the send rate normalised to 1 TS is ¾ Ø. If all four ACs are used, then there is one RAO per two frames. Thus, Ø. The MCCH may only occupy the first TS of a frame. Further RAOs can be added by introducing up to three Secondary Control Channels (SCCH), each occupying one TS per frame. If all TSs of a frame are dedicated to random access, then the total arrival rate is divided by the two RAOs of each TS: Ø ¾. Due to Eqs. 9 and 12, the maximum number of random accesses can be carried out at the same time with a minimum repeat rate and a maximum access frame. Figs. 2 and 3 depict the mean packet delay and the mean throughput Ë in the RF case depending on the number of active MSs with the minimum RF repeat rate ÑÒ ½ ½ ÌË. In the airport scenario [5], the mean waiting time for connection setups is set to Û s. This corresponds to a maximum packet delay s ¾¾ TS. In case of other scenarios as stated in [5], the mean waiting time is Û ½¼ s and thus the maximum packet delay ½¼ s ¼ TS. The values specified in Fig. 2 are the maximum numbers of MSs which can compete for transmission capacity at the same time without exceeding the maximum packet delay s.

In the DF case, an MS has to wait until the beginning of the next access frame before uniformly choosing one RAO. Due to the fact that a random access can only occur after Ï Ì Ã TSs (let us assume that ¼), the arrival rates Ø, ¾ Ø and Ø ¾ normalised to 1 TS have to be adapted accordingly. In Figs. 4 and 5 the maximum number of MSs are specified for the maximum packet delays s and ½¼ s in the DF case. Mean packet delay [TS] 100 75 50 25 8 Ø 2 Ø Ø /2 ½ 0 200 400 600 Æ mobile stations Figure 2: Mean packet delay Æ µ in case of an rolling access frame ¾ Mean throughput Ë [ 1 /TS] 0,4 0,3 0,2 0,1 0,0 8 Ø 2 Ø Ø /2 0 100 200 300 Æ mobile stations Figure 5: Mean throughput Ë Æ µ in case of a discrete access frame V. WAITING PROBABILITY AND WAITING TIME INSIDE TETRA SYSTEMS TETRA systems are waiting systems. A Markov chain model, see Fig. 6, of a waiting system with Å servers (TSs) and Æ clients (MSs) [3] is used in order to calculate the waiting probability Ô Û of an MS, i.e. the probability of newly arrived jobs to find all servers busy, and the corresponding mean waiting time Û per MS. Due to the three Mean throughput Ë [ 1 /TS] Mean packet delay [TS] 0.4 0.3 0.2 0.1 0.0 8 Ø 2 Ø Ø /2 0 200 400 600 Æ mobile stations Figure 3: Mean throughput Ë Æ µ in case of an rolling access frame 800 600 400 200 0 8 Ø 2 Ø Ø /2 ½¼ ½½ ½ 0 100 200 300 Æ mobile stations ¾½ ¾¾ ¾ Figure 4: Mean packet delay Æ µ in case of a discrete access frame N λ t (N 1) λt ( N M +2) λt ( N M +1) λt ( N M ) λt λt 0 1 2 M 2 M 1 M M +1 N 1 N εt 2εt (M 1) εt M εt Figure 6: State diagram of a TETRA system in case of reserved access types of services [5], there are three different service rates. The weighted sum of these service rates, the total service rate Ø, is: M εt M εt Ø Ñ Ñ µ Ø (15) The probability Ô to be in state is with ¾ Æ ¼ ¼ Æ [3] Ô Ô ¼ Ø Ø Æ Ô ¼ Ø Ø Æ ; ¼ Å ½ Å Å Å ; Å Æ (16) With the help of the above equation and È Æ ¼ Ô ½, the state probability Ô ¼ can be determined as follows: Ô ¼ ½ Æ Å Å ½ ½ Ø Ø Æ Ø Ø Æ Å Å Å ½ (17) The waiting probability Ô Û of one MS consists of the probibilities of those states, in which newly arrived jobs encounter all servers busy. The current number of jobs in the waiting queue results from the state number reduced by the

number of servers. The mean length of the waiting queue Æ Õ is the sum of all waiting jobs weighted by the respective state probabilities: Ô Û Æ Å Ô Æ Õ Æ Å ½ Å µ Ô (18) In accordance with LITTLE s result [3], the mean waiting time Û per MS is the sum of all queue lengths divided by the resp. arrival rates: Û Æ Å ½ Å µ Ô Æ ½µ Ø (19) VI. ESTIMATION OF THE SERVICE TIME The service time is defined to be the complete time nescessary for the transmission of a job. Speech transmissions are realised with the help of circuitswitched connections, i.e. one TS is reserved exclusively to the speech connection during the duration of the conversation. This is why the mean service time ¾¼ s of speech connections corresponds to this mean duration [5]. Owing to [1], packets with a maximum size of 133 byte = 1064 bit are to be transmitted over connectionless Basic Links (BL), one of the two V+D Logical Link Control (LLC) link types. The mean packet size of short data messages is expected to be ½¼¼ byte ¼¼ bit. A BL packet is split into one leading Medium Access Control (MAC) ACCESS Protocol Data Unit (PDU) (net payload 56 bit) and Ò subsequent MAC-FRAG PDUs (net payload 264 bit). This is ¼¼ bit bit why Ò ¾ bit for 100 byte packets. If the BS is able to provide uplink resources to the requested transmission of a short data message, then this transmission is carried out within one multiframe. A multiframe consists of 18 frames [1]. The first 17 of them can be used for data transmissions. The BS may assign Ò TSs uniformly distributed over one multiframe to the requesting MS. Not more than one TS per frame is reserved to BL transmissions. Thus, the minimum service time of short data messages is ÑÒ ¾ frames ½ TS TS ½¾ ms (20) Analogously, the maximum service time is ÑÜ ½ frames ½ TS TS ¾¼ ms (21) and as a consequence of this the mean service time of 100 byte packets is: ÑÜ ÑÒ ¾ TS ¾ ½ ms (22) Before transmitting a middle data message, an Advanced Link (AL) connection has to be established, the second V+D LLC link type. The Quality of Service (QoS) parameters AL throughput and LLC segment size have a direct impact on the service time and are negotiated with the help of an AL-SETUP PDU when the connection is established. The throughput can be varied between the maximum available bit rate and 1 /32 of it. The segment size Ð may lie in the range ¾ ¼ byte. Besides an AL-SETUP PDU and an AL-DISC PDU needed for establishing resp. disconnecting an AL connection, Ò AL-Data AL-DATA PDUs and one AL-FINAL PDU have to be transmitted in case of middle data messages (length: Ð Ñ ¾ ¼ byte ½ bit). The Protocol Control Information (PCI) of AL-DATA frames amounts to Ð AL-DATA ½ bit. Moreover note, that AL-FINAL frames contain an additional 32 bit Frame Check Sequence (FCS) and, thus, its PCI amounts to Ð AL-FINAL bit. In the MAC sublayer, the LLC frames AL-DATA and AL-FINAL are mapped to one leading MAC-DATA PDU (net payload Ð MAC-DATA ¾ ½ bit) and several subsequent MAC-FRAG PDUs (net payload Ð MAC-FRAG ¾ bit). In case of maximum segment size Ð ÑÜ = 4 096 byte, one middle data packet with size Ð Ñ can be completely transmitted in one AL-FINAL frame (Ò AL-DATA ¼). This is why the minimum number of TSs Ò ÑÑÒ needed for the AL transmission of an Ð Ñ -size packet results in Ò ÑÑÒ ¾ ÐÑ Ð AL-FINAL Ð MAC-DATA Ð MAC-FRAG TS (23) In case of minimum segment size Ð ÑÒ ¾ byte, one middle data packet with size Ð Ñ is split into Ò AL-DATA ½ segments with Ò AL-DATA ÐÑ Ð ÑÒ ½ TS (24) Each segment is mapped to one MAC-DATA PDU and one MAC-FRAG PDU. Consequently, the maximum number of TSs needed, Ò ÑÑÜ, yields to Ò ÑÑÜ ¾ ¾ Ò AL-DATA ½µ ½ ¼ TS (25) If maximum throughput has been negotiated during connection setup, one TS of the first 17 frames of a multiframe is reserved to the AL connection. The minimum service time ÑÑÒ is with regard to the minimum number of TS Ò ÑÑÒ ÑÑÒ multiframes ½¾ frames ½ TS ¾ TS s (26) The maximum and mean service times, ÑÑÜ and Ñ, can be calculated analogously. The possible results are specified in Tab. 1 under variation of the throughput. The mean net bitrate Ú Ñ Ð Ñ Ñ is also specified in the table on the next page regarding the transmission of middle data messages. The only reason not to always use the maximum segment length is due to the fact that the whole segment has to be retransmitted in case of an erroneous transmission. As a consequence of this, the service time would dynamically outgrow the one corresponding to small segment lengths. The V+D protocol stack does not provide a method to selectively repeat erroneously transmitted MAC PDUs, the LLC frames are mapped to.

Table 1: Service times of AL connections under variation of the QoS parameters throughput and segment size Through- ÑÑÒ ÑÑÜ Ñ Ú Ñ put [TS] [s] [TS] [s] [TS] [s] [bit/s] 1 265 3.754 545 7.721 405 5.738 2 855.600 1/2 565 8.004 1 165 16.504 865 12.254 1 337.033 1/4 1 093 15.484 2 325 32.938 1 709 24.211 676.717 1/8 2 261 32.031 4 669 66.144 3 465 49.088 333.771 1/16 4 597 65.124 9 349 132.444 6 973 98.784 165.857 1/32 9 205 130.404 18 709 265.044 13 957 197.724 82.863 VII. MAXIMUM NUMBER OF USERS IN REGARD TO RESERVED ACCESS Fig. 7 depicts the waiting probability Ô Û depending on the number of active users Æ under variation of the number of servers Å. The arrival rate Ø and the service rate Ø have been used according to Eqs. 14 and 15. 1.0 0.8 0.6 0.4 0.2 Waiting probability ÔÛ 0.0 0 300 600 900 1 200 1 500 Æ mobile stations Å = 3 7 11 15 19 23 27 31 Figure 7: Waiting probability Ô Û depending on the number of active users in case of Å 3, 7, 11, 15, 19, 23, 27 or 31 servers Furthermore note, that one MCCH and one AC have been assumed. The carrier frequency with the MCCH provides three servers, i.e. three time slots, per frame for reserved access. Every additional frequency adds four further servers. This is why the frequency demand yields to Ò Å ½ (27) The number of blocked and aborted connections weightedly contribute to the Grade of Service (GoS) [2]. The airport scenario defines GoS = 5 % [5]. Moreover, GoS = 3 % has been proposed for other scenarios mostly characterized by a lower traffic load. Regarding Ô Û ± resp. 5 %, the number of MSs capacity for reserved access has been assigned to marks the top limit of MSs in case of a GoS corresponding to Ô Û. These top limits of active users are specified in Fig. 8, which is an extract of Fig. 7 for Ô Û ±. According to Eq. 19, Fig. 9 pictures the mean waiting time Û per MS in case of the airport scenario depending on the number of active users Æ. Considering the mean packet delays of this scenario, s ¾¾ TS for Û s and ½¼ s ¼ TS for Û ½¼ s, the maximum number of active users Æ is specified in Fig. 9 corresponding to the resp. mean waiting times Û. Additionally, the maximum number of users is specified in case of the requested V+D connection setup time Û ¼¼ ms ¾½ TS [1]. 6 % 4 % 2 % 0 % Waiting probability ÔÛ Å = 3 7 11 15 19 23 27 31 ¾ ½¼ ½¾ ¾ ¾¾ 0 200 400 600 800 1 000 Æ mobile stations Figure 8: Waiting probability Ô Û depending on the number of active users in case of Å 3, 7, 11, 15, 19, 23, 27 or 31 servers (extract) ¼½ ¼ ¾ VIII. CONCLUSIONS In this paper a Markovian model of an S-ALOHA system with finite population has been developed in order to determine the mean throughput and the mean packet delay in a typical traffic load situation. Considering predefined boundaries of the packet delay, the maximum number of active users has been calculated in case of random access. In general, the random access method RF is to be prefered over the DF method. Mean waiting time Û [TS] 800 600 400 200 0 ½¾ Å 3 7 11 15 19 23 27 31 ½½¼ ½ ¾ ½ ¼½ ½ ¾ ½ 0 300 600 900 1 200 1 500 Æ mobile stations ½¼¾¾ ½¼ ½¼¼ ½½ ½¾½ ½½ ½¾ ½ ½ ¾¼ 11.33 8.50 5.67 2.83 0.00 Figure 9: Mean waiting time Û depending on the number of active users in case of Å 3, 7, 11, 15, 19, 23, 27 or 31 servers Furthermore, another Markovian state model has been presented. This model estimates the waiting probability of new jobs in the system and provides the maximum number of active users which can be served in case of reserved access and a given GoS. Mean waiting time Û [s]

In case of one MCCH, one AC and a typical traffic load, up to Æ = 556 MSs can be served by one BS at the same time consuming Ò = 6 frequencies with Å = 23 servers, i.e. time slots. If no speech connections are requested and only packet data is to be transmitted, then the maximum number of active users is only limited by the random access for Ò ¾. IX. REFERENCES [1] Radio Equipment and Systems (RES) 06, Terrestrial Trunked Radio Voice plus Data, ETS 300-392, European Telecommunications Standards Institute (ETSI), Aug. 1995. [2] B. Walke, Mobilfunknetze und ihre Protokolle. Teubner Verlag, Jan. 1998. [3] L. Kleinrock, Queueing Systems, vol. 1 a. 2. Jon Wiley & Sons, Jan. 1975. [4] S. Lam, Packet Switching in a Multi-Access Broadcast Channel with Applications to Satellite Communication in a Computer Network. PhD thesis, University of California, Los Angeles, 1974. [5] RES 06, Scenarios for Comparison of Technical Proposals for MDTRS, Working Document (91) 23, ETSI, June 1991. [6] Waterloo Maple Inc., Maple V Programming Guide for Release 5, 3rd ed., Nov. 1998.