COMPARISON OF TOKEN HOLDING TIME STRATEGIES FOR A STATIC TOKEN PASSING BUS. M.E. Ulug

Transcription

1 COMPARISON OF TOKEN HOLDING TIME STRATEGIES FOR A STATIC TOKEN PASSING BUS M.E. Ulug General Electric Corporate Research and Developent Schenectady, New York 1245 ABSTRACT Waiting ties have been calculated for an explicit token passing static LAN with two different types of token holding tie strategies. The first strategy allows each station to transit only one inforation packet during a token holding tie and the second allows each station to epty its buffer when it receives the token. It is shown that for a given server utilization both strategies result in exactly the sae ean token rotation tie. The strategy that allows each station to epty its buffer when it gets the token results in approxiately three ties saller ean waiting ties at 50% bus loading. This iproveent in waiting tie becoes greater as the bus loading is increased. Regardless of which of the two strategies is used, the ean waiting tie is independent of the packet length distribution. gated infinite server and find the ean arrival rate of the custoers at the gate. SYSTEM MODEL, Consider an M station syste. Let us assue that on the average there are N stations that are active. Let the stations receive packets at a Poisson rate of G/IM packets per second and let the distribution of the packet lengths be either deterinistic or exponential. We use two different syste odels for two different token holding tie algoriths. For the single packet transission algorith, we assign a certain portion of the total channel bandwidth to a station and calculate the waiting ties assuing that this station can be odeled as an M/D/1 syste. For the ultipacket transission algorith, we odel a station as an infinite server separated fro the custoers by a gate. We assue that every tie the station gets the token the gate opens for a short period of tie and provides bulk service to the waiting custoers. INTRODUCTION In a token passing bus all the stations for a logical ring SINGLE PACKET TRANSMISSION ALGORITHM and the token is passed fro station to station along the ring. Consider an M station syste with total channel capacity The systes with such logical rings are called static because of C bits per second. Let us assue that on the average N all the stations are peranently included in the token passing stations are active. It is expected that, for N<M, the process. bandwidth C' allocated to each of the N stations will have Explicit token passing bus systes with any stations soe general distribution with a ean of suffer fro large overheads and long waiting ties. This is because a great deal of tie is wasted by passing tokens to C'= - ceb stations that are not active. Moreover, in real-tie systes N the stations are allowed to transit only one inforation where Eb is the bandwidth efficiency 'of the syste and is packet per token rotation in order to keep the bus access de- given by lays bounded. The efficiency of the syste iproves when each station is allowed to epty its buffer when it gets the N(x + r) E& (2) token. N(x + r) + (M- N)r In this paper we present the calculation of the waiting ties for the static token passing LANs in order to copare where the perforance of systes using these two different types of token holding strategies. x = ean service tie of inforation packets having fixed lengths and using the channel capacity C. Soe exaples of the calculation of waiting ties and bus access delays are given in References 1, 2 and 4. In one r = total token passing tie including propagation and of our approaches, as in Reference 4, we obtain the syste processing delays. equations by allocating a certain portion of the channel capa- Note that the nuerator of Equation (1) is the usable city to each active station as a function of the packet arrival portion of the channel capacity. rate to the syste and observing the waiting ties experi- Substituting for Eb in Equation (1) Jio Equation (2) we enced by individual stations. In another approach, we use a have the following relationship: CH2106/84/0000/007$ IEEE 7

2 C(x + r ) C' = (Nx + Mr) where T, is the token rotation tie and is given by the following equation: T, = (Nx + Mr) (4) The token holding tie, TI,, is given by the following expression: Ti, = zx + r where z = 1 for the single packet transission algorith and can be as large as required for the ultipacket transission algorith. DELAY ANALYSIS For a station receiving packets in its user ports at a Poisson rate of (G/M), the probability of not receiving a packet during a token rotation tie is (exp-(g/m) T,), and the probability of receiving one or ore packets is (1-exp- (G/M) T,). Because of this, the following relationship holds between N, M, and T,: -_ GT, N=M(I-e M, (6) Because a Poisson arrival process has been assued it is not possible for N to equal M unless K or T, is infinity. Let us now consider N stations receiving one or ore packets during a token rotation. These N stations can transit only one of the packets they received in this rotation. The stations that have received two packets have to wait until the next rotation for the second packet. The stations that have received three packets will have to wait for yet another rotation for the third packet. As a result, on the average there are a lot ore stations transitting packets during a token rotation than just N. Let the ean nuber of stations transitting packets during a token rotation be N'. In order to take this fact into account we odify Equation (6) as follows: where (5) NI= MP (7) This is the probability of receiving one or ore packets during a token rotation plus the probability of receiving two packets plus the probability of receiving three packets, plus.... N' ust equal to the ean nuber of packets that arrive at the user ports of the syste during a token rotation. Therefore, we have the following relationship: NI= GT, (8) By substituting T, in Equation (8) fro Equation (4) we obtain the following result: N' - GMr (9) 1 - GX By substituting N' fro Equation (9) for N in Equation (4) we obtain the following iportant result: Mr T, = ~ 1 - GX (10) This equation enables us to deterine T, without knowing N'. Let us now define the station server utilization, ps, as follows: Ps = (G/M)(x + r) C C' Note that r is greater than lr/c. However, r - I,/C cannot be used for packet transission. Therefore, it has been assued that (x + r) is the total transission tie for the station. Hence, by substituting T, fro Equation () into Equation (11) we have the following relationship: (12) By substituting for N' fro Equation (8) we have the following relationship: Ps = N' CALCULATION OF WAITING TIME The analysis presented in this paper is based on the operation of an active station that has been allocated a bandwidth C' as a certain portion of the total bandwidth of the bus. The station is allowed to transit one inforation and one token passing packet using the bandwidth C' during a token rotation tie. It has been assued that the packets having fixed lengths arrive at the user ports of this station according to the Poisson statistics. Under these assuptions, the waiting tie equation for the station under discussion will be the sae as that of the M/G/1 syste. The type of general distribution will depend on the distribution of the service tie. Since fixed packet lengths were assued, the distribution of service tie will be the sae as that of (l/c'). The distribution of C' is related to the distribution of T,, which is given by Equation (4). Fro the exaination of Equation (41, it would appear as if the statistics of the token rotation tie were dependent on the distribution of x, the service tie. In actual fact, there are N packet transissions during a token rotation tie. Because of this, the distribution of T, depends on the N fold convolution of x, which is an Erlang N distribution. It is well known that for large values of N this distribution is alost deterinistic. Also fro Equations (7) and (8) the token rotation tie is a function of the ratio (G/M). The effect of the fluctuations in the packet arrival rate is once again reduced by the inherently convoluting properties of the logical ring. The results of the coputer siulation indicate that the standard deviation of the token rotation tie is very sall. Because of this, as a good approxiation we use the waiting tie equation of the M/D/1 syste fro Reference as follows: Ps C(x + r ) W= 20 -PSI C' by substituting for C' fro Equation () W= 2(1 Ps - P J T, (14) 8

3 Hence at 50 /o bus utilization the waiting tie of the ultiserver syste is T,/2. By substituting for ps fro Equation (1) we have the following relationship: W= N 2(M- N ) (16) Note that the waiting tie given by Equation (15) is the waiting tie observed in the ultiserver syste of the odel. In order to ake the waiting tie of the odel the sae as that of the token passing bus we would have to add to it the service tie observed in the odel and subtract fro it the actual service tie as follows: XTr w,= w+-- x+r X Since x >> r, the second ter is approxiately equal to T, and the third ter is very sall. Fro Equation (17) it can be seen that at 50% bus loading the waiting tie of the syste is (/2) T,, since W = Tr/2. In the analysis presented fixed packet lengths were assued. If, however, the packet lengths were exponentially distributed, the waiting tie equation would not change since the distribution of the token rotation tie is approxiately independent of the distribution of the packet lengths. The coputer siulation of a token passing bus indicates that there is very little difference between perforance of a syste using fixed packet lengths and exponentially distributed packet lengths with the sae ean. This is given below. PACKET LENGTHS FIXED (205 &sec) I EXPONENTIAL Token Rotation Tie ws I s I I s I 2.109, , , , Let the ean nuber of packets received and transitted by a station during a token rotation be s and let the ean nuber active stations as in the case of the single packet transission algorith be denoted by N. Then we have the following relationship: s= [$ITr (18) Note that in Equation (18) we used ((GIN) instead of (G/M). This is because in soe rotations the packets arrive at the rate of (GIN) and in soe others there are no arrivals, resulting in an average arrival rate cif (GIM). Based on this definition of s we can rewrite Equation (4) as T, = Nsx + Nr (19) Reeber that the token rotation tie for the algorith that allows the transission of only one inforation packet is T, = N x + Mr (20) By substituting for s in Equation (19) l ro Equation (18) and by substituting for N in Equation (20) fro Equation (8), it can be shown that T, given by Equations (19) and (20) are equal and the following relationship holds: -= s N (21) Substituting for s in Equation (19) fro Equation) and solving for T, we obtain the following result: Mr T, = ~ 1 - GX (22) The ean nuber of active stations on the ring can now be obtained fro the following equation using the value T, obtained fro Equation (22): N=l-e ~- G Tr The waiting tie is calculated as follows: W=Q+ (s - l)(x + r) 2 where Q is the ean queueing delay during a token rotation. As will be shown later, Q in Equation (24) is relatively sall in coparison to the waiting tie given in Equation (17). Moreover, Q does not increase rapidly with the increasing values of server utilization. This is because no packet has to wait ore than one token rotation to receive service. The second ter in Equation (24) represents the ean waiting tie after the station receives the token. Since on the average there are s packets in the service area, the waiting tie inside the gate is given by the following equation: (s - 1) s 1 (s -.-.(x + r) l ) ( x ~ = 2 2 S In other words, the su of integers fro zero to (s - 1) averaged over s ultiplied with the service tie (x + r) is the ean waiting tie inside the gate. This is because the first packet does not wait at all, the second packet waits (x + r), the third packet waits 2 (x + r), the fourth packet waits (x + r), etc. 9

4 Typically, the waiting tie given by Equation (24) is approxiately equal to one-half of the token rotation tie even under heavy loads. This is the advantage of allowing unliited token holding tie so that a station can transit all of its packets. The disadvantage is that the bus access tie is now bounded at a higher level. In order to calculate Q the following queueing odel will be considered for an individual station of the static ring. Consider an infinite server syste or a single server having an infinitely large channel capacity separated fro the custoers by a gate. Let the custoers arrive at a Poisson rate of (C/M) packets per second and the gate be opened for a very short period of tie when the station receives the token, i.e., every T, seconds. When the gate is opened all the custoers are aditted to service and the gate is closed. Service takes only (sx + r) seconds. However, the gate is not opened again until the station receives the token. Based on this odel Q is defined as the ean waiting tie to the gate opening. Note that during a token rotation a given station receives either one or ore packets (i.e., the rotation is busy), or no packets at all (i.e., rotation is idle). For the idle rotations, there are no packet arrivals and no delays. Although the ean packet arrival rate is (GIM), for the busy rotations the actual arrival rate is (GIN). Clearly, the delays should be calculated only for the busy rotation and the packet arrival rate (GIN) should be used. Let us assue that there are no packets left in the syste when the gate is closed. If there is only one packet arrival, the distribution of the arrival tie will be exponential with a ean of (N/G). If there are two arrivals, the distribution of the second arrival tie will be Erlang 2 with ean of 2(N/G). If there are three arrivals, the distribution of the third arrival will be Erlang with ean of (N/G), etc. If we assue that the token rotation tie is infinitely large, then the ean arrival tie after the gate closure, TA,, is obtained by ultiplying these arrival ties with the corresponding Poisson probabilities and adding as follows: TA, = ( (N/ G) Prob (one arrival) + 2 (N/G) Prob (two arrivals) + (N/G) Prob (three arrivals +... ) / (1 - e G T, -- (25) Since the probability of having zero arrivals does not appear in the nuerator of Equation 25, it is necessary to divide all the probability ters by the ter (1 - exp- (GIN) T,) so that the weights will add up to unity. Since the token rotation tie is not infinitely large, it is necessary to calculate the eans of the gaa, exponential and Erlang distributions, that fall within the interval fro zero to T,. Because these distributions are truncated, in calculating the eans it is necessary to noralize the distributions by dividing the by the area under the gaa (a) curve fro zero to T,. This area, because of the special relationship between gaa and the Poisson distributions, is equal to the probability of receiving (a) or ore packets in T, seconds. (26) where A = (G/N) and T = T,. The first ter in Equation (26) is the noralizer for the Poisson probability weights. The first ter inside the (a) su is the Poisson probability weight. The nuerator of the next ter is the ean of the gaa (a) function in the interval zero to Tr. The ter in the denoinator is the area under the gaa (a) fro zero to T,. The denoinator is the noralizer for the truncated ean given in the nuerator. The exact solution of Equation (26) is very difficult to obtain. However, it is possible to obtain a closed for solution if one akes the following approxiation: T 21" s e-"p- 'dr (29) 0 a(1 + eat) When this approxiation is used Equation (26) reduces as shown below: The results obtained fro siulation of the waiting tie indicate that this is a very good approxiation. In coputation of Equation (26) the following wellknown result is used to calculate the area under the curve between zero and T, of the gaa (a) function: (1) The truncated ean of the gaa (a) function is calculated for different values of (a) and shown below: a=l a=2 a= a=4 a=5 1 - e-*'[ T + i] - A +- () (4) -- e-at 4T XT4 4h2T I 12AT2] (5) A [ A!!! -- 5 e-at ST h4t5 5XT4 x [ x 4! 4! +- 2ox2T2 4! + ""I1 4! 40

5 a=n _- n A e-~7 When TA, is coputed using the above listed equations, the ean waiting tie to the gate opening, Q, is calculated by subtracting TA, fro Tr as follows: Q = Tr - TA, (8) By substituting Q obtained fro Equation (8) in Equation (24) the waiting tie is obtained. If one chooses to use the approxiate solution of Equation (26) the waiting tie equation results in the following: (9) The outputs of Equations (26) and (9) do not differ ore than 4% and surprisingly enough the approxiate result produces a closer correlation with the results of siulation. Since the second ter of Equation (9) is very sall, the waiting tie of the ultipacket transission case is approxiately equal to Tr/2 independent of bus loading. On the other hand, with the help of Equation (17) it was shown that at 50% bus loading the waiting tie of the single packet transission case was (/2)Tr. Hence, at 50% bus loading the waiting tie of the ultipacket algorith is three ties saller than that of the single packet algorith. This iproveent in waiting tie is obtained by giving each station an unliited aount of tie to epty its buffer. In order to keep the waiting tie tie bounded, it is necessary to ipose a tie liit on the duration of the ultipacket transission. This liit can best be set up based on the token rotation tie observed by each station. The reason for selecting the token rotation tie rather than the token holding tie is as follows: The token holding tie is a paraeter that is identically assigned to all stations. Yet not all stations are active at the sae tie. A bound based on token holding tie applies to the case when all stations are active. However, this is very rare for a syste having the Poisson arrival statistics. Therefore, a liit based on this rare event will be too restrictive ost of the tie. The token rotation tie, on the other hand, is a syste variable. A liit set on the token rotation tie is valid regardless of how any stations are active. Multipacket transission algorith used with a liit based on the token rotation tie as observed by the individual stations retains ost of the benefits resulting fro this approach while aintaining a bound on the waiting tie. PRIORITY SCHEMES It is shown that for a given server utilization both the single and the ultipacket strategies result in exactly the sae ean token rotation tie. The ultipacket strategy that allows each station to epty its buffer when it gets the token results in saller waiting ties. This is due to the fact that soe stations have ore packets to transit and soe stations have less. In other words, there are peaks and valleys in the utilization of the bus by the individual stations. One of the priority schees suggested for the token passing LAN systes is as follows: The station receiving the token checks the rotation tie. If the rotation tie is saller than a predeterined value, the station is able to transit lower priority packets. If not, the station can only transit a first priority packet. Such a priority schee has the effect of equalizing the output of all stations by filling the valleys with lower priority packets. In other words, the fluctuations that the ultipacket transission algorith takes advantage of are now eliinated. Cobination of the priority schee described above with the ultipacket transission algorith is very likely to result in an increased variance of the token rotation tie. This in turn will increase the waiting ties for the first priority packets beyond what would have been experienced if there were no second priority packets. RESULTS Two sets of coputational results have been obtained. The first set is for the token holding tie algorith which allows only one inforation packet transission per token rotation. The second set is for the algorith which allows the stations to epty their buffers when they receive the token. Both sets of coputations are repeated foir different values of the token passing tie Y. These results have been obtained for systes having 50 and 250 stations connected to a 5 Mb/s explicit token passing bus. In the calculations 20- and 128-byt'e packet sizes have been assued for the token and the inforation packets, respectively. Figure 1 shows how the waiting tie of a syste having 50 stations and using the single packet transission algorith increases with the increasing packet arrival or traffic rate for four different token passing tie!;. Figure 2 provides the sae inforation for a syste having 250 stations. Note that in this case larger delays have been observed for saller traffic rate & I I I I to Troffic G IN [pkts/sec/stotionl Figure 1. Waiting tie versus traffic for stalic ring, single packet transission: 50 stations. 41

6 1001 Token Poesing (us),,, Q b I a % Ifs t E Traffic G/M Ipkts/sec/stotionl Figure 2. Waiting tie versus traffic for static ring, single packet transission; 250 stations. Figure shows how the waiting tie of a syste having 50 stations and using the ultipacket transission algorith increases with the increasing packet arrival or traffic rate for four different token passing ties. Figure 4 provides the sae inforation for a syste having 250 stations. Once again the increased nuber of stations result in larger delays for saller traffic rate. Figure 5 shows a coparison of the waiting ties for two systes having 50 stations each but using different token holding tie strategies. The syste using the ultipacket transission algorith has uch saller waiting ties than the one using the single packet transission algorith. These curves are plotted for a token passing tie of 175,US. Figure 6 provides the sae coparison for two systes having 250 stations each. - 0, 10- E a- Y i-' 6- C.J,.> 4- l I I I I I Traffic G/M [pkts/secjstotion) Figure. Waiting tie versus traffic for static ring, ultipacket transission; 50 stations. 10- a- 0 2 i I2 Traffic G/M Ipkts/sec/station) Figure 4. Waiting tie versus traffic for static ring, ultipacket transission; 250 stations..f.-i lo0l Legend * p-s+lo 0 a f I I I I I Traffic G/M Ipkts/sec/stationJ Figure 5. Waiting tie versus traffic for static ring, 50 stations, single and ultiple packet transission. Figure 7 shows how the token rotation tie of a syste having 50 stations and using the single packet transission algorith increases with the increasing traffic rate for different values of token passing ties. Figure 8 provides the sae inforation for a syste having 250 stations. Figures 7 and 8 also represent variations of token rotation tie with the traffic rate for 50 and 250 station systes using the ultipacket transission algorith since both strategies result in the sae token rotation tie. Figure 9 shows how the ean nuber of active stations in a syste having 50 stations and using the single packet transission algorith increases with the increasing traffic rate for different values of token passing ties. Figure 10 provides the sae inforation for a syste having 250 stations. The ean nuber of active stations for the case of ultipacket transission can be obtained fro these graphs by siply dividing N by s where s is given by Equation (18). 42

7 Troffic G/M Ipkts/sec/stotionl Figure 6. Waiting tie versus traffic for static ring, 250 stations, single and ultiple packet transission. I I I 7----[ t I2 Troffic G/M Ipkts/~ec/st.otionl Figure8. Taken rotation tie versus trafflc ffor 250 stations, single packet transission. z - "1 - I I 1 I I I to Troffic G/H [pkts/sec/stat.ion) c.> 4 I) tk 4-0 L a, a E z c 0 z 10 - ~ 0 li LO M 40 so 611 Traffic G/M Ipkts/sec/statianl Figure 7. Taken rotation tie versus traffic for 50 stations, single packet transission. Rgure9. Nversus traffic for 50 stations, single packet transission. Figures 11 and 12 show how the syste utilization varies with the traffic rate for systes having 50 and 250 stations. Note that this variation is independent of the type of token holding tie algorith used. The results fro the coputer siulation will be presented in a separate paper. CONCLUSIONS Waiting ties have been calculated for an explicit token passing static LAN with two different types of token holding tie strategies. The first strategy allows each station to transit only one inforation packet during a token holding tie, and the second allows each station to epty its buffer when it gets the token. For a given server utilization both strategies result in exactly the sae ean token rotation tie. The strategy that allows each station to epty its buffer when it gets the token results in approxiately three ties saller waiting ties. This iproveent becoes greater as the bus loading is increased. However, in this case the bus access delays are not necessarily bounded. Multipacket transission algorith used with a liit based on the token rotation tie as observed by the individual stations retains ost of the benefits resulting fro this approach while aintaining a bound on the waiting tie. The use of a decentralized priority schee based on token rotation tie is likely to reduce the advantages of the strategy that allows the stations to epty their buffers when they receive the token and thereby results in saller delays. 4

8 P C 0.J U N 0.4-.J 2.A Token PossLng (us) 0 as2i,-,- A X%* nd.---, Traffic<G/M [pkts/sec/stationl Figurelo. N versus traffic for 250 stations, single packet transission. 0.0 I I I I I a 10 TraffLc G/M (pkts/sec/station) Figure 12. Utilization versus traffic for 250 stations, single packet transission. of the one that allows the transission of one inforation packet per token rotation tie. The siulation results also show that the waiting tie is independent of the distribution of the packet length for both strategies. ACKNOWLEDGMENTS The author would like to express his thanks to Mr. C.R. Stein, Dr. N.R. Shapiro, Miss M.R. Laliberte and Dr. T. Sayda for their help and advice. U io Traffic G/M [pkts/sec/stationl Figure 11. Utilization versus traffic for SO stations, single packet transission. The results of the coputer siulation indicate that the standard deviation of the token rotation tie for the strategy that allows each station to epty its buffer is larger than that REFERENCES A.G. Konhei and B. Meister, Waiting Lines and Ties in a Syste with Polling, JACM 21, pp , July M.E. Ulug, G.M. White and W.J. Adas, Bidirectional Token Flow Syste, 7th Data Counications Syposiu 11, (41, Mexico City, October L. Kleinrock, Queueing Systes, Vol. I, Theory, Wiley- Interscience, New York, M.E. Uiug, Calculation of Waiting Ties for a Real Tie Token Passing Bus, Proc. of Coputer Nehvorking Syposiu, Silver Spring, Maryland, Deceber