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Analysis of an asymmetric polling system
Computers Ops Res. Vol. 24, No. 4, pp. 317-333, 1997
Pergamon
PH: S0305--0548(96)00057-3
O 1997 Elsevier Science Lid Printed in Great Britain. All rights reserved 0305-0548/97 $17.00+0.00
ANALYSIS OF A N ASYMMETRIC POLLING SYSTEM M. Khalidt:~", P. D. Vyavahare§ b and H. B. Kekrdl ¢ 'Department of Computer Science and Engineering, Indian Institute of Technology, Bombay, 400 076, India ~Department of Electronics and Telecommunication Engineering, S.G.S. Institute of Technology and Science, 23 Park Road, Indore, 452 003, India ~l)epartment of Computer Engineering, Thadomal Shahani Engineering College, Linking Road, Bandra (West), Bombay, 400 050, India
(Received August 1995; in revised form July 1996) Scope and Purpese---Polling models have been used extensively to analyze the performance of a variety of systems. Their range of applications include computer communication, robotics, traffic and transportation, assembly work of cars etc. One of the major reasons for their numerous applications is that the cyclic allocation of a resource (server) to multiple clients is quite natural and fair to these fields of engineering. Analysis of these models can be used for performance evaluation and its improvement, system comparison and optimization. In their applications to computer systems, polling models have been used to study scheduling of moving arms of the secondary storage devices, data transfer from multi-drop lines to central computer and load sharing in multiprocessor computers. In the present context, polling models are used to analyze local area networks. In this article, an analysis is developed for a reverse round robin polling system in which the sequence of polling is reversed from round to round. Expressions for major performance measure of the system as a function of load are then derived. The analysis developed in this article is numerically evaluated for a high speed local area network, and is compared with that of a round robin system, as an illustration. The work has relevance in study of other domains in which single resource is shared by multiple queues in either round robin or reverse round robin fashion. Abstract--Polling models are used extensively in the field of operations research to analyze the performance of a variety of systems that can he viewed as a distributed queuing system served by a single server and to evaluate their operational characteristics. In the context of computer communication, a Local Area Network (LAN) with controlled access on a single transmission media can he viewed as such a polling system in which queues are being served in cyclic order. In this article, a LAN with reverse round robin scheduling for channel arbitration, called M-net [1], is modeled as a discrete time Markov chain representing an asymmetric polling system. Expressions for its performance measures like average channel utilization, mean and variance of delay and blocking probability at a station are derived. For a system of M stations, the solution complexity is of the order of 2 ~M- ~. The analysis approach is applicable to both round robin and reverse round robin systems. Finally, performance measures of such a system are compared with that of Expressnet, as a representative example of round robin system. It is found that at lower arrival rates, the performance of M-net is comparable to that of Expressnet. However, with increasing values of arrival rates, M-net outperforms Expressnet. The analysis presented is applicable to the class of systems in operations research that can be viewed as multiple single buffer queues served by a server in round robin or reverse round robin fashion. © 1997 Elsevier Science Ltd, All rights reserved t To whom correspondence should be addressed (e-mail: [email protected]). ~: Dr. M. Khalid is currently an Associate Professor in the Dept. of Computer Science and Engineering, Indian Institute of Technology, Bombay. He received B.Sc. degree in Electrical Engineering from Aligarh Muslim University, Aligarh, India; M.Tech. degree in Electrical Engineering from Indian Institute of Technology, Delhi; and Ph.D. degree in ComputerCommunication Networks from Indian Institute of Technology, Bombay in 1974, 1977 and 1982 respectively. He joined the faculty of Computer Science and Engineering Dept., HI" Bombay in 1981. His teaching and research interests include Computer-Communication Networks, Switch Architecture for Networking and Performance Evaluation of Networking and Switching systems. He has published over thirty articles in the above areas. § Dr. P. D. Vyav~are is a reader in the Depm'tment of Electronics and Telecommunication Engg., S. G. S. Institute of Technology and Science, Indore 452 003 (M.P.). He received B.Sc. and M.Sc. (Physics) degrees from University of Indore, Indore in 1972 and 1974 respectively and M. Tech. in Electrical Engineering from liT Bombay in 1976. He worked in Tam Institute of Fundamental Research from 1976 to 1982 as a communication engineer. Since 1982 he is working with S. G. S. Institute of Technology and Science Indore. He was research scholar in the Department of Computer Science and Engg., liT Bombay on deputation from his parent institution under QIP - Ph.D. program of MHRD, Govt. of India from 1990 to 1994 and received his Ph.D. degree in 1995 in the field of high speed local area network. His research areas include access schemes for high speed LANs and their performance evaluation and telecommunication networks. He has published over ten articles. ~[ Dr. H. B. Kekre is Professor and Head in the Department of Computer Engg. at Thadomal Shabani Engineering College, Bombay, India. He was formerly professor in the Department of Computer Science and Engg. at Indian Institute of Technology, Bombay. He received the B.E. Hons. degree in Telecommunication Engineering from Jabalpur University, Jabalpur, India, in 1958 and the M.Tech. in Electrical Engineering from liT, Bombay, India, in 1960. He received the M.Sc. degree in Electrical Engineering from University of Ottawa, Ottawa, Canada, in 1965 and Ph.D. from IIT, Bombay in 1970. Since 1960 he has been on the Faculty of Electrical Engg. at Irr, Bombay. In 1974, he joined the Computer Center, fiT, Bombay and from 1979 to 1985 he was Head of the Department of Computer Science and Engineering at f i t Bombay, From 1963 to 1965 he was at the University of Ottawa, Ottawa, Canada, under a Commonwealth scholarship programme. His fields of interest are simulation, picture processing, system identification and computer communications. He has published over 100 journal articles on these topics. He is Member of the Computer Society of India, the Indian Society for Theory of Probability and its Applications, Senior Member of the IEEE and Fellow of IETE, India. 317
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M. Khalid et al. I. INTRODUCTION
In recent decades, generic polling models, with single server attending multiple queues in cyclic order, have been used to analyze the performance of a variety of systems including production, robotics and computer communications [2]. The analysis developed on these models is of interest for performance evaluation and improvement of the systems. With the advent of computer communication networks, an extensive study has been carried out on polling systems for data transfer among computer and terminals and recently between computers on a LAN [3]. In this article, an asymmetric polling system model is developed for a Unidirectional Bus System (UBS) LAN, called M-net [1]. M-net belongs to the class of implicit token based UBS LANs that include Expressnet [4], Fasnet [5], Buzznet [6], TokenLess Protocols (TLP) [7], X-net [8], Z-net [8] and FBC-net [9]. These LANs are robust as compared to explicit token schemes which require provisions for handling abnormal situations like loss or duplication of token. M-net has Expressnet like simplicity and is based on folded bus UBS architecture with two outbound channels and one inbound channel [I,10]. The MAC protocol of the system employs reverse round robin scheduling to provide higher channel utilization and improved delay performance. A comparative study [8,11 ] reveals that the performance of X-net and TLP-3 is better than many other systems. It is found that performance bounds of M-net [1] are comparable to that of X-net and TLP-3 but it is architecturally simpler and more reliable than these networks. Since M-net, X-net, TLP-3 and FBC-net employ reverse round robin scheduling, they may be viewed as an asymmetric polling system. As the modeling complexity of an asymmetric polling system is more than that of a symmetric polling system, only simulation studies of X-net, TLP-3 and FBC-net have been reported [7-9]. In this article, performance of M-net is studied analytically. It is modeled as a discrete time Markov chain. Symmetric round robin system, in which arrivals at each of the M stations are statistically identical, has been analyzed and solved efficiently using polling system model [5,12]. Takagi [13] has analyzed symmetric polling systems with single buffer assuming general probability distributions of transmission time and walk time. His analysis requires solving 2 ~ linear simultaneous equations to calculate mean delay at a station. Recently, analysis for asymmetric round robin polling systems has been reported by Takine [14] which also needs solving 2 Mequations. The solution complexity of the model presented in this article is 2 2¢M-'>. The analysis approach is applicable to both round robin and reverse round robin systems. In Section 2, architecture and MAC protoco! are presented [1]. Model development of M-net is reported in Section 3. In Section 4, expressions for various performance measures of M-net are deduced. This is followed by computation of numerical results and their comparison with that of Expressnet in Section 5. Finally, the article is concluded in Section 6. 2. ARCHITECTURE AND MAC PROTOCOL OF M-NET The architecture of M-net is exemplified in Fig. 1. It consists of a folded unidirectional bus having three sections namely OUTbound channel 1 (OUTI), OUTbound channel 2 (OUT2) and an INbound channel (IN). A station is connected to OUT1, OUT2 and IN through a set of directional passive taps (S 1, $2, T1, T2 and R). A station senses the status (i.e. carrier activity caused by upstream transmission) of OUT1 and OUT2 through the sense taps S1 and $2 which are upstream T1 and T2 respectively. It transmits signals through transmit tap TI (T2) on OUT1 (OUT2), depending on the status of the outbound channel and receives signals from the IN through receive tap R. The signal received may be either a DU destined for the station or an event occurring on the IN. The MAC protocol relies on the presence or absence of the carder as sensed by taps S I, $2 and R. To facilitate its explanation, functions and events, representing significant activities on OUT1, OUT2 and IN at a station, are defined in generic sense as follows. A Boolean function C(t,.) (where • is OUT1 or OUT2 or IN) takes a value 1 when the presence of carder is detected on the corresponding channel at time t and is 0 otherwise. Beginning Of Carrier BOC(.) and End Of Carrier EOC(.) on the channel occur when C(t,-) takes a transition from 0 to 1 and from 1 to 0 respectively. If td is the carrier detection time and detection circuit is close to the tap, then the detection of events BOC(.) and EOC(.) is delayed by td seconds with respect to the actual beginning and end of the carder at the sensing/receiving tap. A succession of transmissions of all DUs, that are separated by the carrier detection time td, constitutes a train. A Boolean function TRAIN (t,.), on the
Analysis of an asymmetric polling system
319
TRAIN(t,.) = C(t - t a,.) + C(t,.)
( 1)
channel, is defined as
Events Beginning Of Train BOT(.) and End Of Train EOT(.), representing beginning and end of a train on the channel, occur when TRAIN (t,*) makes transitions from 0 to 1 and 1 to 0 respectively. Events EOC(OUT1), EOC(OUT2), and EOT(IN) are known as synchronizing events and are used for possible transmissions by a station. A station is said to be alive if it is monitoring synchronizing events successfully and is participating in the process of their assured presence. A station which is not alive is said to be dead. An alive station with a DU to transmit is known as backlogged. It is termed as idle otherwise. The network is said to be alive if at least one of the stations is alive. The network which is not alive is said to be dead. The access protocol executed at a station is as explained below. A station monitors simultaneously OUT1, OUT2 and IN channels. If the carder activity is in progress on OUTI channel, the station waits till EOC(OUT1) occurs. At this instant, if backlogged, it starts its DU transmission and simultaneously monitors activity on OUT1 for possible transmission from upstream. If R
R
Inbound channel (in)
R
D
/'2 S'~
--
SI TI1
R¸
Outbound channel 2 (out 2) /2 $2
72 S~
S
(oat l) o
.
°
.
•
•
°
o
o
•
°
•
S •
•
( a ) Stations in a line
( b ) station I and M ¢olocnted
Fig. 1. Architectureof M-net. Og~ 24-4-B
_
siTII) lf /lll
320
M. Khalid et al.
such a transmission is detected, within td seconds of the transmission, it aborts its own transmission in favour of the upstream one and waits for next EOC(OUTI). It then repeats at every such EOC(OUT1) the above steps. If no transmission from upstream is detected in first td seconds, the station continues its DU transmission. The station thus completes its transmission successfully on OUT1 appending its DU to the end of the partial train. After the transmission of its DU in the train, the station does not sense EOC(OUT1) in the current round and hence will not be able to transmit another DU on OUT1. If the backlogged station observes that the carrier activity is not in progress on OUTI but is progressing on OUT2, it waits for EOC(OUT2). It then attempts transmission of the DU on OUT2. The procedure followed for transmission on OUT2 is the same as that for OUT1. If the station does not observe carrier activities on either of the outbound channels then the station waits till the occurrence of EOT(IN). At this instant, it transmits a special signal for td seconds on OUT1, called locomotive [4], to ensure existence of synchronizing events. It then attempts, if backlogged, access on OUT1. Note that in M-net the end-to-end propagation delay on each channel is same and taps S1 and R of any two stations are connected to the channels in such a way that propagation delay between their S 1 taps is same as that between their R taps. Therefore, the event EOT(IN), used by all stations as the synchronizing event to start a new train, reaches any station exactly at the same time as the beginning of the new train on OUT1 channel because of a station which is upstream on OUTI. Hence, locomotives of all stations overlap on the channel. Following the locomotive transmission, the most upstream backlogged station on OUT1 completes its transmission successfully and the normal MAC procedure follows. The complete train formed consequently is seen on IN by all the stations. If ~'is end-to-end propagation delay on a channel then the propagation delay between S1 and R taps at any given station is 2r. Therefore, the inter-train gap, defined as the time between the end of the last DU in a train on IN channel and the beginning of the next train on it, is (27 +2td) seconds. Thus, stations with indices 1 - m -< M get access opportunities in the order 1 to M on OUTI and then in the order M to 1 on OUT2 in every train. A train, as seen on IN, followed by the inter-train gap constitutes a cycle. Figure 2 shows typical sensing and transmission activities on OUT1 and OUT2 channels and receiving activities on IN channel at a station of index i3. It is assumed that transmission time of a DU, T, is constant. Stations i I and i2 are upstream and station/4 is downstream with reference to station i3 on OUT1. The diagram shows a typical situation when all these stations are backlogged before the end of the partial train and hence transmit a DU on OUT1 as well as on OUT2 in a train. Note that since sensing tap Sm ($2) is upstream the transmit tap T~ (Te) on OUT1 (OUT2), the transmission of i 3 o n OUT1 (OUT2) is not sensed by S~ ($2). Figure 3 shows the corresponding space-time diagram indicating activities on OUT1, OUT2 and IN channels. The EOC(IN) occurs t# seconds after the actual end of transmission on IN channel at a station. The locomotive of the next train is transmitted by the station on the occurrence of EOT(IN) i.e. if no further activity is sensed in next td seconds after EOC(IN). Therefore, the time interval between the actual end of transmission on IN channel at the R tap of station and the start of locomotive on OUT1 by it is 2td seconds. In an alive network, the presence of a synchronizing event is always detected on IN channel in a time interval of (2~" +2td) seconds. If a station (after it becomes alive) does not detect it for (21" +2t#) seconds on IN, it concludes that the network is dead. It then initiates a process of regenerating synchronizing events through a cold-start procedure. This procedure is similar to the one which is followed in Expressnet [4]. The state diagram for the M-net access protocol at a station is shown in Fig. 4. Initially, a station is in dead state. When the request for making it alive is generated, the station starts monitoring carrier activities on all the three channels. If the carrier activity is detected within (2~" +2td) seconds then it is confirmed that the network is alive. In such a case, station follows the normal procedure of DU transmissions and generation of synchronizing events. If no activity is detected on any one of the channels for (27" +2td) seconds then the station confirms that the network is dead. In such a case, it executes cold-start procedure on OUT1 to generate synchronizing event EOT(IN). The presence of a DU at an alive station is indicated by a variable DU_buffer which takes values full or empty depending on the presence or absence of the DU for transmission at the station. As the attempt and defer mechanism on OUTI and OUT2 channel is same, a variable called OUT_CH is defined that takes values OUT1 or OUT2 depending on the outbound channel on which DU transmission may be attempted by the station. If the station, on becoming alive, detects that the network is alive, then it waits for a synchronizing event. At that instant, it moves to appropriate state. However, if the station detects its presence on
Analysis of an asymmetric polling system
321
outbound as well as on inbound channels, it synchronizes with the synchronizing event of an outbound channel. If the carder activity is detected on both the outbound channels, then it synchronizes with EOC(OUT1) for possible DU transmission or for moving to appropriate state. When the station becomes alive, by executing the cold-start procedure, it waits for EOT(IN). At this instant it transmits the locomotive to ensure presence of EOT(IN), even if no station has a DU for transmission in the current train and goes to ready state. The station then attempts (and defers if necessary) the DU transmission on OUT1 if the state of DU_buffer is full at the instant of its moving to ready state. Otherwise, it goes in defer & wait state and returns to ready state if its DU_buffer is full at EOC(OUT_CH). If OUT_CH is OUT1 then the station attempts for DU transmission on O U T I when its DU_buffer is full and it is in ready state. If no transmission from upstream is detected on the outbound channel, then it completes its transmission, sets OUT_CH to OUT2 and moves to defer & wait state. Otherwise, it aborts the transmission, moves to defer & wait state and waits for next EOC(OUT1). If the station does not have a DU till the train has passed on OUT1, and if OUT_CH=OUT1, then the station changes OUT_CH to OUT2. It then waits for EOC(OUT2) and undertakes possible DU transmission on OUT2 following the same procedure as that for the transmission on OUT1. On detection of EOT(OUT2) or after the completion o f DU transmission on OUT2, the station moves to wait state for the start of the next train and the procedure is repeated till the station is alive.
iI
C
12
(t, O U T 1)1 0
2 Transmit 1 tap on out I
iI
Time
EOC (OUT 1) BOC (OUT 1) EOC (OUT l) i2
i3
•
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.
.
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Time
C (t, out 2) j . ~ E O C (OUT2)
Prop delay between $I and $2 +ta
Time
i
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C(t.
P
Time
in) ,
q
~--
Time
I
fl Locomotive Train (t, in)
II
i~
Cycle
--I • ~---2~ + ta - - ~
Time BOT (in) EOT (in) Fig. 2. Activity profiles on OUTI. OUT2 and IN channels with respect to station i3 when stations i, i2, i3 and /4 are backloggedat both the polling instants in a cycle under consideration.
J,
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M. Khalid et al. 3. M O D E L D E V E L O P M E N T OF M - N E T
As explained in Section 2, a server polls stations first in order 1 to M on OUT1 and then in the order M to 1 on OUT2 in a polling cycleJ It serves the station for time T, if it has a DU present in its buffer at its polling instant, before moving to the next station. There are, therefore, 2M polling instants in a polling cycle. Thus, it is modelled as a reverse round robin polling system with the following assumptions: (1) The system consists of M single buffer stations, each of which can hold a fixed length DU. The transmission time of a DU is T seconds. (2) The propagation delay, ~',,, between station m and (m+ l) (1 < - m -< ( M - 1)) is constant. (3) DUs arrive at a station m ( l -< m -< M) with Poisson arrival rate Amand arrival of a DU at a station is independent of arrivals at other stations. (4) DUs arriving at an occupied buffer (backlogged station) or during the service (transmission) of a DU are lost (blocked). A station at its polling instant is either backlogged (station state l) or idle (station state 0). In order to study the performance of M-net, system state is defined as a set of station states at their respective polling instants in a polling cycle. Since the state of a station depends only on its inter-visit time, the state of the system exhibits Markovian property. M-net, therefore, may be modeled as a discrete time Markov chain with a state space of size 2 TM. However, if respective propagation delays between two transmit taps of station M on OUT1 and OUT2 and between the two taps of station 1 on OUT2 and IN are negligibly small, stations 1 and M will have utmost one DU per cycle in steady state. But, station 1 may have DUs when server polls it on OUT1 and OUT2 in the first cycle. Therefore, maximum number of DUs in the first cycle is 2 ( M - 1) + 1 and is L = 2 ( M - 1) in subsequent cycles. Hence, there are 2 L recurrent states.
c~ i2 i3 i4
i4
i3 O
i2
iI
il .= ~2 i3 i4
Time Fig. 3. Space-time diagram of M-net corresponding to Fig. 2. i A polling cycle is defined as the inter-visit time to the same point in the configuration. The convention of starting a cycle from polling instant of station 1 is followed here for convenience.
Analysis of an asymmetric polling system
323
Therefore, the size of the state space, S, of such a system under steady state is 2 L and the dimension of the single step transition probability matrix, P, is 2L× 2 L. ThUS, State of the system may be represented by an L-tuple (sl, s2..... s/) of (0,1) elements. An element in the L-tuple corresponds to the state of a particular station when the server polls it on a particular outbound channel. Let the starting point of the cycle be polling instant of station 1 on OUT2. The cycle, in this case starts when station 1 is polled on OLrI'2 and ends when station 1 is revisited on OUT2. Then, the first element of the L-tuple, sz, corresponds to state of station 1 at its polling instant on OUT2. The element si at i 'h Rosition, for 2--- i -< M, of the tuple indicates state of the i 'h station when the server polls it on OUT1. For (M +1) --< i --<2(M- 1), si at the i 'h position in the tuple indicates state of the ( 2 M - i) 'h station when polled on OUT2. 3.1. Composition of transition probability matrix Let I and J be the system states corresponding to two consecutive polling cycles respectively. An element P(I,J) of single step transition probability matrix P is a product of L terms. Each term, Pi(l,J), 1-< i <- L, depends only on the inter-visit time at the corresponding station. Inter-visit time at a station is a sum of two terms. First term is the walk time of the server between the
C Station dead
'~
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Network dead
(
Station alive
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Network alive
Monitor C (t, out 1 ) = 0 & C (t, out 2) - 0
I~
CI
q
EOT (in)
~ E O C ~ ( o u1), t (out--CH~
=f
Ready
(transmit
I
locomotive, out = CH: - out 1)
[
Train (t, out 1) = 0 & EOC (out 2) (out-CH: = out 2)
DU buffer = empty -
)
~
(o
] DU_buffer : f u l ~ ~ y [ (start DU TX. on "~totf.
(t-t~,out_OH) - 0 (complete D U TX.)
[ EOC (out_CH) [ DU_buffer = ] empty
C
out_CH - out I &
train (t, out 1) - 0 out_CH = out I out 2)
(out_CH: out_OH = out 2
(Transmit) EOT (out 2) Fig. 4. State diagram of M-net access scheme.
(out-CH: = out 2)
M. Khalid et al.
324
consecutive visits to the station. The second term corresponds to the transmission time incurred in serving the DUs at the backlogged stations during the consecutive visits. Walk-time o f the server between the M-I
consecutive visits to rn th station (1<- m <-- ( M - 1)) on OUT1 first and then on OUT2 is '6m,M)=2 E
k=m
r~. Similarly, walk-time o f the server between visits to station m (2 < - m <- M) on OUT2 and the next visit m-I
to the same station on OUT1 is "6m,1)=2 £
r k. Walk-time for end stations is 2r. The number o f DUs,
k=l
K, served during the inter-visit time at a station corresponding to i 'h polling instant, ri(K), depends on the system states I and J corresponding to two consecutive polling cycles. The inter-visit time ~(K) is, therefore, given by
2r+KT
for i= 1 and 0-
"6i,1)+KT
2(M- lj j=2
sfl)<--2M- 3
for 2<-i<--Mand 2 ( M - 1)
~(/0 =
£
0
i- 1
sfl)+ £ sj(J)<--2i- 3
j = ( 2 M - i+ 1) -
(2)
j=l
"62M - i,M) + KT for M+ 1<--i.~2(M- 1) and 0
~
s~(J)--<2(i- M) - 1
j=t2M-i+l)
-
As can be seen from equation (2), r/(K) for a given value o f K is station index dependent. Moreover, the maximum value o f K is also a function o f i and M. Thus, the i 'h term o f P(I,J), P~(I,J), is given by
Pi(l,j)= {
e-a'~'~x)
if si(J)= 0
1 - e - a'~
(3)
Note that for (M + l) -< i < - 2 ( M - 1), Ai=AEU_i, since each station is polled twice in a cycle by the server. Therefore, an element P(I,J) can be written as 2 ( M - I)
P(I,J)=
I1 [e-a,',~x).6(si(J))+(1-e-a,',~x~).si(J)]
(4)
i=l
Where S(.) is a Dirac delta function.
3.2. Steady state probabilities Let p(d) be the steady state probability of a system state and p be the associated steady state probability vector o f size 2 L. The system state probabilities can be determined from
p [ l - P] =0
(5)
where I is the identity matrix and 0 is a null vector o f dimensions (2 L × 2 L) and 2 L respectively.The necessary and sufficient condition for equation (5) to have a non-trivial solution is that rank of [I - P] should be less than 2 L [15]. It is shown in Appendix A that the rank o f [ I - P ] i f ( 2 L - 1). As the rank o f [I - P] is exactly one less than the number o f elements in p, solution o f equation (5) will give ratios o f steady state probabilities. Applying normalization relationship on sum o f the probabilities, steady state probabilities can be evaluated from these ratios. The procedure for finding steady state probabilities thus consists o f performing linear operations on rows o f [I - P] to get an upper triangular matrix. If these operations do not change the value o f the determinant o f the resultant matrix, then the errors due to numerical computations are reduced. 2 It may be noted that for M= 2 and a~ = As = A, M-net with stations at the two ends o f OUT 1 degenerates to corresponding Expressnet. Its state probabilities as adopted from [5] with M = 2 are: X4
Pt°'°)= 1 - x2 +x2y
(6)
2One way to achieve this is by using Gauss elimination technique. An alternate way to evaluate an element of the upper triangular matrix directly from [! - P] is to take ratios of the values of two determinants of the appropriate matrices of dimensions (1,/) and (1- I, 1- 1) [16]. However,this approach, in general, takes more computational time.
Analysis of an asymmetric polling system
325
p~o.l)=p.,o)=(x -2_ l).p~0.0)
(7)
p..i)=p
(8)
where x = e ~ and y = e at.It can be shown that equations (6)-(8), satisfyequation (5) verifying the validity of the analysis approach, proposed above, for M=2. 4. P E R F O R M A N C E
MEASURES
OF M-NET
Major performance measures of a network are channel utilization,mean and variance of delay, and the blocking probability. In the following subsections methods for calculating these measures are explained. 4.1. M e a n channel utilization Let pN(n) be the probability mass function (pmf) of the number of DUs, N, served in a polling cycle. Then pN(n) is given by L
p~(n)= Y~ p(J).8( ~ s , ( J ) - n) J~S
(9)
i= I
The average number of DUs served in a cycle, E(N), is given by 2 ( M - I)
E(N)=
Y~ np~(n)
(10)
n=0
The train on inbound channel is followed by an inter-train gap of (2~" +2ta) seconds. Normally, td is a very small fraction of T and hence can be neglected in calculating the channel utilization. Therefore, the mean channel utilization, CM, of the network is given by E(N) T Cu= E(N)T+2r -
1 2a I+--
(11)
E(N)
where t~= r/T. Using this relationship, mean channel utilization profile of M-net under different values of a and arrival rates can be obtained. 4.2. Mean and variance o f delay at a station As every station is polled twice in a cycle, DUs arriving at a station may be served on either of the outbound channels. All intermediate stations may transmit none, one or two DUs in a cycle. However, end stations will have utmost one DU per cycle because the time latency between the server's departure instant from OUT2 (OUTI) and the subsequent polling instant on OUT1 (OUT2) for station 1 (M) is assumed to be negligibly small. A DU at a station experiences delay because of channel access waiting time and its transmission time on it. Mean and variance of the delay in M-net, even for identical arrival rates at all stations, are station index dependent because the respective inter-station walk times from OUT1 to OUT2 and from OUT2 to OUT1 in a polling cycle are station index dependent. The probability, p(si= 1, K), that K number of DUs will be served during the inter-visit time ~,.(K) and si= 1 is E W
2(Mp(Jll)p(l)8( j=2
1)S)(1)
- IO
YJ~,(J)= 1
f o r i= 1 and O<--K<--2M- 3 p( Jll)p( l) 3(X j.2u _ <,_ , ~s,(I ) + X j,, , s,(J) - l O W
¥,/t*j(,/)= 1
p(si= 1,K)=. for 2<-i<-M and O<-K<-2i - 3
x
~¢J1$~1)= I
(12)
p(J38(X~2~,-.-l:j(J)-g)
f o r (M+ 1)--
326
M. Khalid et al. 2M - 3
~, p(si=l,h')
fori=l
K=0 2i-3
~, p(s~= 1,K) p(si= 1)=' x.o
for 2<-i<-M
(13)
2(i- hi0- 1
E
p(s~=l,K) for M+ l < i < 2 ( M - 1)
K=0
and the conditional probability, p(Klsi = 1), that K number of DUs are served during the station inter-visit time that has elapsed prior to i 'h polling instant, given that the corresponding station is backlogged at its polling instant, is p(Klsi= 1)= p(si= l, . p(si= 1)
1<-i<-2(M - 1)
(14)
Where K is a non-negative integer such that (2M- 3 for i=1 K<~2i- 3 for 2<--i<--M 12(i - M) - 1 for (M + 1)--
(15)
The probability density function, fw,(wlK), of the access waiting time W~of a DU served at i 'h polling instant for a given r,(K) is [17] A,e ~'w fw~(wIK) = ~ e A ~ 1 0
O<--w< ~i(K)
(16)
otherwise
Where ~,(K) is given by equation (2). The conditional mean and second moment of W,, for 1<- i -<2(M- 1), are then given by E(WilK)= 7/(K)ea'~'
1 hi
(17)
and
l
E(W21K)= e ~ ' ' ~ - 1
[ r2(K)ea,T,(to 2~(K)ea'"<~ 2e~'"¢~° ,~22 ] ,~i + A~
(18)
Note that as A,.---, 0% right hand sides of equations (17) and (18) reduce to ~(/Q and r2(K), the inter-visit time and square of the inter-visit time at i ,h polling instant respectively, and the delay expression matches with the bounds derived in [1]. Similarly, as At---* 0, equations (17) and (18) reduce to ~ g ~ and ~'Ttt~) 3 I respectively and equation (16) reduces to a uniform probability density function ~ for 0 <- w - %(K) and is zero otherwise. The mean and second moment of waiting time, E(W~) and E(W~i), for ( 1 - i -<2(M- 1)) are given by E(W,)= E E(W, IK)p(KIs,= 1)
(19)
VK
and E(W2)= ~ E(W~IK)p(KIs~= 1)
(20)
VK
Where K is a non-negative integer satisfying equation (15) Let to~.,, and to~.mbe the waiting time at a station m, when a DU is served on OUT1 and OUT2 respectively. Then, their mean values are given by E( toij,)= E(Wm) for 2<-m<-M and
(21)
Analysis of an asymmetric polling system
327
f E(W.~) for m = 1 E(oh~,,)= / E(W~u_.) for 2<-m<-(M- 1)
(22)
A DU may be served on OUT1 or OUT2. Therefore, the mean, E(to.), of the waiting time, to=, at a station m (1-- m --< M), is expressed as
E( o~.,,,) E( w,,,)=
for m = 1
p(s,. = 1) p(s2M-.= 1) p(s,,,= 1)+p(sz~_,.= 1) E(w.j.)+ p(sm= 1)+p(s2M-,.= 1) E(o~.) for 2<--m<--(M- 1) (23)
for m =M
E(toj.m)
Similarly, expressions for second moments of the waiting times. E(tozt.,.).E(w~.m) and E(to~.), can be deduced. Let D~.m and D2.,. be the normalized delay at a station m, normalized with respect to DU transmission time T. when a DU is served on OUT1 and OUT2 respectively. Then, the mean of Dt..~ and D2.mare given by (24)
E(Dl.m)= T [E(wLm)+T]for 2<-m<-M and 1
E(D2~.)= ~ [E(toz~)+ T] for l <-m<-M - 1
(25)
The mean, E(Dm), of the normalized delay at a station m, D,., (1~ m --< Mr) is
E(Dz.m) E(D,.) =
for m= 1
p ( s m = 1) p(S2M-,~= 1) p(sm = I ) +p(s2u-.,= 1) E(D,~)+ p(sm= 1)+p(s2M-~,= 1) E(D2~) for 2<--m<--(M- 1) (26)
E(Dw.)
for m=M
The variance of the normalized delay at a station m, o~o~,(1-< m -< M) can then be easily expressed as 2 _ O"w.2 ' _-O'Dm--
~ I [E(a,m2 ) _ E2(wm)]
(27)
4.3. Blocking probability at a station When a DU arrives at a station, its buffer gets full. DUs generated during the time interval in which the buffer at a station is not empty are lost (blocked). Therefore, the blocking probability (or probability of loss) at a station is another important performance measure in a single buffer system. The blocking probability at a station served at the i ~ polling instant is [17] pb(i)=
XI(E(W,)+D 1 + X~(E(W3+ r )
for 1-
(28)
A DU stored in the buffer may be served on either of the outbound channels. Therefore, the blocking Table I. End-to-end distance and DU size for various values of a ot
End*to-end distance (Kin)
Number of bits in a DU
0. I 1.0 I0 100
0.2 2.0 20.0 20.0
1000 1000 I000 100
328
M. Khalid et al.
No. of stations (/14)= 5 end-to--end prop. delay
a 0.80 -
/
/
packet transmission time •
a=O.l
/ f
/
t
/ f a = l
M-net -- Expressnet
O
.,.
e"
~
o.40-
~
1
a=10
a= 100 0.00
I
I0
I
I
[
,
I
I O0
1000
10,000
100,000
Lambda (arrivals/s) Fig. 5. Mean channel utilization versus arrival rate for M-net and Expressnet with M=5 and or=0.1, I, 10 and 100.
probability at a station with station index m, pB(m), is given by Am(E( W~) + T)
f o r m= l and M
I + A~(E(W.~)+ T)
pB(m) =
A~(E(WI)+ T)
I + a,.(E(Wm)+ T) +
×
(29)
p(Sm = 1)
p(Sm= 1) +p(szAc_m= 1)
Am(E(W~_ m)+ T) p ( s ~ _ m= 1) × 1 + Am(E(W2M_m)+T) p(sm= 1)+p(Sm-m= 1)
for 2<--m<--M- 1
5. NUMERICAL COMPUTATIONS AND THEIR INTERPRETATION The analysis approach presented in Section 3 and 4 is illustrated here. The performance of M-net and Expressnet (as a representative example of round robin system) with equally spaced stations and identical arrival rates, A, are evaluated for this purpose. Results are obtained for four different values of a i.e. a=0.1, 1, 10 and 100. Table 1 shows end-to-end distance on one of the channels and number of bits in a DU for various values of a. The propagation delay and transmission rate are assumed to be 5 microseconds/Km and 100 Mbps respectively. DU size is 1000 bits for a=0.1, 1 and 10. For a = 100, however, DU size is 100 bits. Alternatively, one could have assumed a constant DU size of 1000 bits for all values of a. However, for a transmission rate of 100 Mbps, this would mean end-to-end distance of 200 Kin at or= 100. Figure 5 shows variation of mean channel utilization versus Poisson arrival rate A with a as a parameter for a system of M= 5 stations. At lower values of A, difference in channel utilization of M-net and Expressnet is insignificant. For example, the performance of the two systems is almost same for values of A up to 1000. However, as A increases, M-net has better channel utilization than the corresponding Expressnet system. This is because in M-net, at lower values of A, the probability with which a station transmits on both channels in a cycle is very small. However, as A increases, the
Analysis of an asymmetric polling system
329
probability with which a station utilizes both access opportunities in a cycle increases. Finally, at very large values of A, the channel utilization, CM, of M-net and, Ce, of Expressnet saturate to values corresponding to the situation in which stations are backlogged at all the polling instants in most of the cycles. Thus, as A----, 0% CM---* Cu(max) and C e ---* Ce(max) where CM(max)=l/[1 + a / ( M - 1 ) ] and Ce(max) = 1/[ 1 + 2a/M] are the bounds on channel utilization in M-net and Expressnet respectively Ill. Figure 6 shows the variation of normalized mean delay at stations with A for a system of M= 5 stations and a = 10. Delay performance at all stations in symmetric Expressnet is same. However, mean delay at a station, in M-net, is station index dependent. In M-net, delay performance E(DI)=E(D2,1)at station 1 and E(Ds)=E(Dj.5) at station 5 are same. At lower values of A, mean delay at end stations in M-net are same as that for Expressnet. This is because, at these values of A, the inter-visit time at end stations in M-net and Expressnet are comparable. However, at higher values of A, inter-visit times for these stations in M-net are more than that in Expressnet. Therefore, delays experienced at these stations are also more than that in Expressnet. Delays at intermediate stations in M-net are smaller than the delay in Expressnet for all values of A. It can be seen that at station 2, for a given A, mean delay on OUT1, E(DI.2), is less than mean delay on OUT2, E(O2.2). This is because ~-(2,1) < 7-(2,5). For station 3, delays on both outbound channels, E(Dj.3) and E(D2,3), are same and are equal to the mean delay, E(D3). Mean delay at station 3 is less than the mean delays at stations 2 and 4 over a large range of h. It is approximately half of the Expressnet mean delay. This is because walk time to the central station in M-net is half (or close to half if M is an even number) as compared to that in Expressnet. As the arrival rate increases, the intervisit time at a station also increases resulting in increased delay. As A ---. o% mean delays at all stations in M-net, except end stations, are same. This happens because at such high values of A, as soon as the server departs a station, DU arrives at the station with very high probability. In such a case, mean delay at a station approaches the inter-visit time and a DU will be served on either of the outbound channels with equal probability. This observation is also in agreement with the results derived for bounds on 30-No. of s t a t i o n s (M) ffi 5
E (DI), E (D 5)
alpha = 10 f
M-net 25 --
i
_ Expressnet
/
/
""
"-
~
/
/ /
~
~/
E (D2,2), E (/)1,4)
20 ell
it
I
o
15 o N °~ o z 10 E (DI,2), E (Di,4)
5--
0
,
l0
J
i i ,,,,I
,
I00
i
, , Jllll
,
I000
i
i ~ iiitl
I
10,O00
,
, It,Hi
100,000
,
,
, , ,,,,I
i
1,000,O00
Lambda (arrivals/s) Fig. 6. Normalized mean delay on O U T I and O U T 2 at stations and normalized mean delay at stations versus arrival rate for M-net and Exprcssnet with M = 5 and a = 10.
M. Khalid et al.
330
maximum normalized delay in [ 1]. The maximum values of mean delays, on individual channels at intermediate stations, in M-net and mean delay at any station in M-net and Expressnet in Fig. 6, are in exact agreement with the bounds derived in [1]. As per [1], maximum normalized delays at an intermediate station m, on OUT1 and OUT2, ale Dj~(max)= [1(re, l) + 2 ( m - 1)T]/T, and D~(max)= [1(re,M) +2(M-m)T]/Trespectively. The maximum value of mean delay at an intermediate station in M-net is Dm(max)=(a + M - 1). For end stations in M-net, it is 2(a + M - 1) and is (2a + M) for a station in Expressnet. Variation of normalized mean delay with mean channel utilization for a = l0 with M and station index as parameters is shown in Fig. 7. Variations for M=5 arc abstracted from Figs 5 and 6. Similarly, respective profiles for M=3 and 4 are also depicted in the figure. For the same channel utilization, delay performance of M-net is better than that of Expressnet, even at end stations. This is because, for the same channel utilization in both the cases, the arrival rate in Expressnet is higher than that in M-net (see Fig. 5). This results in lower mean inter-DU arrival time for Expressnet than in M-net. In M-net, delay at end stations for M=5 is lower than that for M=3 and 4 except for lower values of channel utilization. This is because for the same channel utilization, A for M=5 is smaller than that for M=3 and 4. Thus, in such a case, the mean DU inter-arrival time for M=5 is greater than that for M=3 and 4. Figure 8 shows the variance of delay versus mean channel utilization for a system of M=5 stations and a = 1 and 10. In M-net, for a given value of a and channel utilization, variance is highest at end stations and lowest at the middle station. This is because of the reduction in the number of stations that the server visits during the inter-visit time for middle stations as compared to the end stations. These stations may or may not have DU at their respective polling instants. At lower values of channel utilization, the variance of delay at a station is non-zero because of randomness between the arrival instant of the DU and the instant at which server polls the station. The variance at a station drops to zero at saturation (i.e. as A --* oo), since a new DU arrives as soon as the server departs the station, transmitting the previous 30 - No. o f s t a t i o n s ( M ) = 3 , 4 , 5 alpha = 10 25 --
m--
Mr5 m=l,5
M-net Expressne!
M=4
'/
M=3]
I
I
m = 1,3
I I
I I I
20
m = 1,4
?/ /
o
.
15 M
es O
ffi4
IM ffi5
M=3 m=2
Z
I0
01 0
I
I
I
I
I
I
0.05
0.10
0.15
0.20
0.25
0.30
Mean channel utilization Fig. 7, N o r m a l i z e d m e a n delay versus m e a n channel utilization for M-net and Expressnet with a = 10 a n d Mffi3,4 a n d 5.
Analysis of an asymmetric polling system
331
DU, and all cycles are of maximum length. Variation in blocking probability at a station with arrival rate A is shown in Fig. 9 for M = 5 and a = 1 and 10. At lower values of ~, the blocking probability at all stations is negligibly small. As the value of A increases, the blocking probability also increases. This increase is higher for a = 10 than for a = 1 caused by a higher cycle time for a = 10. Over a large range of A, blocking probability at end stations in M-net is larger than that in Expressnet. However, at intermediate stations in M-net it is smaller than that in Expressnet. This is as a consequence of delay performances at these stations. 6. CONCLUSIONS
In this article, a model of a reverse round robin polling system is developed for the analysis of a UBS LAN called M-net. It is modeled as a discrete time Markov chain with a state space of size 2 2tM-t) for a system of M stations on the network. It is assumed that stations have a single buffer with Poisson arrivals. It is found that the system is inherently asymmetric in nature because the inter-visit time at a station is a function of station index m. State transition probability matrix approach is used to calculate system state probabilities through the solution of 2 2tM-m)linear simultaneous equations. Expressions for mean channel utilization, mean and variance of delay and blocking probability at a station are derived in terms of these state probabilities. The analysis approach developed in this article can be used to evaluate round robin as well as reverse round robin systems in a unified manner. Finally, the performance of M-net for a system of 3, 4 and 5 stations is numerically evaluated and compared with those of Expressnet as an illustration. Expressnet is taken as a representative example of LANs based on round robin system. It is found that the performance of M-net is better than that of Expressnet at higher values of A and is comparable otherwise. The queuing analysis presented in the article is applicable to a class of engineering systems that can be modelled as a polling system with single buffers and round robin or reverse round robin scheduling. 5O
No. of stations (M) = 5 alpha = 1.0, 10 40
,;Y
2" 30
>
2O
M-net - - -- Expressnet
!
', I I I I I I I
10
I I
0.0
0.2
0.4
0.6
0.8
1.0
Mean channel utilization Fig. 8. N o r m a l i z e d variance o f delay versus m e a n channel utilization for M-net a n d Exprcssnet with a = i, 10
and M=5.
332
M. Khalid et al. 1.0 No. of stations (M) = 5 alpha = 1.0. 10 M-net -- Expressnet 0.8
0.6 es a e~
.o
o
0.4
= i.5 = 2,4 =3 =1
a=lO,
m-- 1,5-
0.2 m
~.
01=
10
100
1000
I 0,000
100,000
1,000,000
Lambda (arrivals/s) Fig. 9. Blocking probability versus arrival rate for M-net and Expressnet with M=5 and a = 1 and 10.
REFERENCES 1. Khalid, M., Vyavahare, P. D. and Kekre, H. B., M-net: Architecture and access protocol. Technical Report TR-132-94, Dept. of Computer Science and Eng., Indian Institute of Technology, Bombay 400 076 INDIA, February 1994. 2. Levy, H. and Sidi, M., Polling systems: applications modelling and optimization. IEEE Trans. Commun. , 1990, 38, 10 1750-1760. 3. Takagi, H., Applications of polling models to computer networks. Comp. Networks and ISDN Sys., 1991, 22, 193-211. 4. Tobagi, E A., Borgonovo, E and Fratta, L., Expressnet-a high performance integrated service local area network. IEEE J. Select. Areas Commu~. 1983, SAC-l, 5 898-913. 5. Fine, M. and Tobagi, E A., Performance of unidirectional broadcast local area networks: Expressnet and Fasnet. IEEE J. Select. Areas Comraun., 1983, SAC-I, 5 913-925. 6. Gerla, M., Rodrigues, P. and Yeh, C., Buzznet: A hybrid random access/virtual token local network. Proc. of the Globecom, 1983, December, 1509-1513. 7. Rodrigues, P., Fratta, L. and Gerla, M., Tokenless protocols for fiber optic local area networks. IEEE J. Select. Areas Commun., 1985, SAC-& 6 928-940. 8. Abeysundara, B. W. and Kamal, A. E., High speed local area networks and their performance--a survey. ACM Computing Survey, 1991, 23, 2 221-264. 9. Economou, E. G., Bolls, S. and Philokyprou, G., FBC-net: a reverse round robin LAN for voice and data traffic. Comp. Commun., 1991, 14, I 44-52. 10. Economou, E. G. and Bolls, S., G-net: Implicit token-passing bus LAN. Electronics letts, 1991, 27, 16 1449-1451. 11. Vyavahare, P. D., Khalid, M. and Kekre, H. B., Recent trends in implicit token based local area networks. Proc. IEEE Int. Conf. on Computer Networks, ICCN-92, 1992, December, FT30.-40. 12. Mack, C., Murphy, T. and Webb, N. L., The efficiency of N machines unidirectionally patrolled by one operative when walking time and repair times are constant. J. Roy. Star. Soc., 1957, 19B, 166-172. 13. Takagi, H., Queueing analysis of polling systems. ACM Computing Surveys, 1988, 20, 5-28. 14. Takine, T., Takahashi, Y. and Hasegawa, T., Exact analysis of asymmetric polling systems with single buffers. IEEE Trans. Commun., 1988, COM-36, 10 1119--1i 26. 15. Bellman, R., Introduction to matrix analysis. McGraw Hill, 1960. 16. Grotschel, M., Lovasz, L. and Schrijver, A. Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, 1980. 17. Kaye, A. R., Analysis of a distributed control loop for data transmission. Proc. Syrup. Comput. Commun. Networks TeletrqO~c, 1972, April, 47-58.
A n a l y s i s o f an a s y m m e t r i c polling s y s t e m
333
APPENDIX A THE RANK OFA STATE TRANSITION MATRIX Theorem 1. Let the single-step transition probability matrix of a L-state irreducible, aperiodic Markov chain be P. Then rank of matrix [ l - P ] is ( L - l). Proof The chain is aperiodic and irreducible, therefore O
(30)
Where P(i,j) is the (i,j)'* element of the transition probability matrix P. Then, the sum of all columns of[P - I] is a null column vector. Therefore, L columns of matrix [P - I] are linearly dependent. Hence, the rank of [P - I] must be (L - 1) or less. Let fl~, ~ ..... tL be scalars, not all of which are zero, such that j=l
t~P(ij) - ti=O f o r all !
(31)
Let t , be a scalar among t l to tL which satisfies the following property likl>-Itjl f o r all i
(32)
Without loss of generality, one can assume that ilk= 1. Consider k th row of the matrix [P - I] which is given by [e(k, 1),P(k,2),...P(k,k) - l,...e(k,L)]
(33)
( t j P ( k , i )+ /$zP(k,2)+ ... + t~P(k,k)+... + tLP(k,L) ) - ik=O
(34)
From equation (3 i) one gets
As t , = 1; ~ P(k,j)= 1; O<]P(i,j)< 1 f o r all 1 <~i,j <- L and t i < 1 f o r all 1< i < L, therefore there is only one solution to equation j=l
(34) given by fl, = ~ . . . . . tk . . . . iL= 1
(35)
Hence k '~ column of matrix [I - P] can be represented by a unique linear combination of the remaining (L - 1) columns. Therefore there are ( L - 1) independent columns in matrix [I - P]. As the column rank of a matrix is same as the row rank, therefore rank of the matrix [I - P] is (L - 1).
Pergamon
PH: S0305--0548(96)00057-3
O 1997 Elsevier Science Lid Printed in Great Britain. All rights reserved 0305-0548/97 $17.00+0.00
ANALYSIS OF A N ASYMMETRIC POLLING SYSTEM M. Khalidt:~", P. D. Vyavahare§ b and H. B. Kekrdl ¢ 'Department of Computer Science and Engineering, Indian Institute of Technology, Bombay, 400 076, India ~Department of Electronics and Telecommunication Engineering, S.G.S. Institute of Technology and Science, 23 Park Road, Indore, 452 003, India ~l)epartment of Computer Engineering, Thadomal Shahani Engineering College, Linking Road, Bandra (West), Bombay, 400 050, India
(Received August 1995; in revised form July 1996) Scope and Purpese---Polling models have been used extensively to analyze the performance of a variety of systems. Their range of applications include computer communication, robotics, traffic and transportation, assembly work of cars etc. One of the major reasons for their numerous applications is that the cyclic allocation of a resource (server) to multiple clients is quite natural and fair to these fields of engineering. Analysis of these models can be used for performance evaluation and its improvement, system comparison and optimization. In their applications to computer systems, polling models have been used to study scheduling of moving arms of the secondary storage devices, data transfer from multi-drop lines to central computer and load sharing in multiprocessor computers. In the present context, polling models are used to analyze local area networks. In this article, an analysis is developed for a reverse round robin polling system in which the sequence of polling is reversed from round to round. Expressions for major performance measure of the system as a function of load are then derived. The analysis developed in this article is numerically evaluated for a high speed local area network, and is compared with that of a round robin system, as an illustration. The work has relevance in study of other domains in which single resource is shared by multiple queues in either round robin or reverse round robin fashion. Abstract--Polling models are used extensively in the field of operations research to analyze the performance of a variety of systems that can he viewed as a distributed queuing system served by a single server and to evaluate their operational characteristics. In the context of computer communication, a Local Area Network (LAN) with controlled access on a single transmission media can he viewed as such a polling system in which queues are being served in cyclic order. In this article, a LAN with reverse round robin scheduling for channel arbitration, called M-net [1], is modeled as a discrete time Markov chain representing an asymmetric polling system. Expressions for its performance measures like average channel utilization, mean and variance of delay and blocking probability at a station are derived. For a system of M stations, the solution complexity is of the order of 2 ~M- ~. The analysis approach is applicable to both round robin and reverse round robin systems. Finally, performance measures of such a system are compared with that of Expressnet, as a representative example of round robin system. It is found that at lower arrival rates, the performance of M-net is comparable to that of Expressnet. However, with increasing values of arrival rates, M-net outperforms Expressnet. The analysis presented is applicable to the class of systems in operations research that can be viewed as multiple single buffer queues served by a server in round robin or reverse round robin fashion. © 1997 Elsevier Science Ltd, All rights reserved t To whom correspondence should be addressed (e-mail: [email protected]). ~: Dr. M. Khalid is currently an Associate Professor in the Dept. of Computer Science and Engineering, Indian Institute of Technology, Bombay. He received B.Sc. degree in Electrical Engineering from Aligarh Muslim University, Aligarh, India; M.Tech. degree in Electrical Engineering from Indian Institute of Technology, Delhi; and Ph.D. degree in ComputerCommunication Networks from Indian Institute of Technology, Bombay in 1974, 1977 and 1982 respectively. He joined the faculty of Computer Science and Engineering Dept., HI" Bombay in 1981. His teaching and research interests include Computer-Communication Networks, Switch Architecture for Networking and Performance Evaluation of Networking and Switching systems. He has published over thirty articles in the above areas. § Dr. P. D. Vyav~are is a reader in the Depm'tment of Electronics and Telecommunication Engg., S. G. S. Institute of Technology and Science, Indore 452 003 (M.P.). He received B.Sc. and M.Sc. (Physics) degrees from University of Indore, Indore in 1972 and 1974 respectively and M. Tech. in Electrical Engineering from liT Bombay in 1976. He worked in Tam Institute of Fundamental Research from 1976 to 1982 as a communication engineer. Since 1982 he is working with S. G. S. Institute of Technology and Science Indore. He was research scholar in the Department of Computer Science and Engg., liT Bombay on deputation from his parent institution under QIP - Ph.D. program of MHRD, Govt. of India from 1990 to 1994 and received his Ph.D. degree in 1995 in the field of high speed local area network. His research areas include access schemes for high speed LANs and their performance evaluation and telecommunication networks. He has published over ten articles. ~[ Dr. H. B. Kekre is Professor and Head in the Department of Computer Engg. at Thadomal Shabani Engineering College, Bombay, India. He was formerly professor in the Department of Computer Science and Engg. at Indian Institute of Technology, Bombay. He received the B.E. Hons. degree in Telecommunication Engineering from Jabalpur University, Jabalpur, India, in 1958 and the M.Tech. in Electrical Engineering from liT, Bombay, India, in 1960. He received the M.Sc. degree in Electrical Engineering from University of Ottawa, Ottawa, Canada, in 1965 and Ph.D. from IIT, Bombay in 1970. Since 1960 he has been on the Faculty of Electrical Engg. at Irr, Bombay. In 1974, he joined the Computer Center, fiT, Bombay and from 1979 to 1985 he was Head of the Department of Computer Science and Engineering at f i t Bombay, From 1963 to 1965 he was at the University of Ottawa, Ottawa, Canada, under a Commonwealth scholarship programme. His fields of interest are simulation, picture processing, system identification and computer communications. He has published over 100 journal articles on these topics. He is Member of the Computer Society of India, the Indian Society for Theory of Probability and its Applications, Senior Member of the IEEE and Fellow of IETE, India. 317
318
M. Khalid et al. I. INTRODUCTION
In recent decades, generic polling models, with single server attending multiple queues in cyclic order, have been used to analyze the performance of a variety of systems including production, robotics and computer communications [2]. The analysis developed on these models is of interest for performance evaluation and improvement of the systems. With the advent of computer communication networks, an extensive study has been carried out on polling systems for data transfer among computer and terminals and recently between computers on a LAN [3]. In this article, an asymmetric polling system model is developed for a Unidirectional Bus System (UBS) LAN, called M-net [1]. M-net belongs to the class of implicit token based UBS LANs that include Expressnet [4], Fasnet [5], Buzznet [6], TokenLess Protocols (TLP) [7], X-net [8], Z-net [8] and FBC-net [9]. These LANs are robust as compared to explicit token schemes which require provisions for handling abnormal situations like loss or duplication of token. M-net has Expressnet like simplicity and is based on folded bus UBS architecture with two outbound channels and one inbound channel [I,10]. The MAC protocol of the system employs reverse round robin scheduling to provide higher channel utilization and improved delay performance. A comparative study [8,11 ] reveals that the performance of X-net and TLP-3 is better than many other systems. It is found that performance bounds of M-net [1] are comparable to that of X-net and TLP-3 but it is architecturally simpler and more reliable than these networks. Since M-net, X-net, TLP-3 and FBC-net employ reverse round robin scheduling, they may be viewed as an asymmetric polling system. As the modeling complexity of an asymmetric polling system is more than that of a symmetric polling system, only simulation studies of X-net, TLP-3 and FBC-net have been reported [7-9]. In this article, performance of M-net is studied analytically. It is modeled as a discrete time Markov chain. Symmetric round robin system, in which arrivals at each of the M stations are statistically identical, has been analyzed and solved efficiently using polling system model [5,12]. Takagi [13] has analyzed symmetric polling systems with single buffer assuming general probability distributions of transmission time and walk time. His analysis requires solving 2 ~ linear simultaneous equations to calculate mean delay at a station. Recently, analysis for asymmetric round robin polling systems has been reported by Takine [14] which also needs solving 2 Mequations. The solution complexity of the model presented in this article is 2 2¢M-'>. The analysis approach is applicable to both round robin and reverse round robin systems. In Section 2, architecture and MAC protoco! are presented [1]. Model development of M-net is reported in Section 3. In Section 4, expressions for various performance measures of M-net are deduced. This is followed by computation of numerical results and their comparison with that of Expressnet in Section 5. Finally, the article is concluded in Section 6. 2. ARCHITECTURE AND MAC PROTOCOL OF M-NET The architecture of M-net is exemplified in Fig. 1. It consists of a folded unidirectional bus having three sections namely OUTbound channel 1 (OUTI), OUTbound channel 2 (OUT2) and an INbound channel (IN). A station is connected to OUT1, OUT2 and IN through a set of directional passive taps (S 1, $2, T1, T2 and R). A station senses the status (i.e. carrier activity caused by upstream transmission) of OUT1 and OUT2 through the sense taps S1 and $2 which are upstream T1 and T2 respectively. It transmits signals through transmit tap TI (T2) on OUT1 (OUT2), depending on the status of the outbound channel and receives signals from the IN through receive tap R. The signal received may be either a DU destined for the station or an event occurring on the IN. The MAC protocol relies on the presence or absence of the carder as sensed by taps S I, $2 and R. To facilitate its explanation, functions and events, representing significant activities on OUT1, OUT2 and IN at a station, are defined in generic sense as follows. A Boolean function C(t,.) (where • is OUT1 or OUT2 or IN) takes a value 1 when the presence of carder is detected on the corresponding channel at time t and is 0 otherwise. Beginning Of Carrier BOC(.) and End Of Carrier EOC(.) on the channel occur when C(t,-) takes a transition from 0 to 1 and from 1 to 0 respectively. If td is the carrier detection time and detection circuit is close to the tap, then the detection of events BOC(.) and EOC(.) is delayed by td seconds with respect to the actual beginning and end of the carder at the sensing/receiving tap. A succession of transmissions of all DUs, that are separated by the carrier detection time td, constitutes a train. A Boolean function TRAIN (t,.), on the
Analysis of an asymmetric polling system
319
TRAIN(t,.) = C(t - t a,.) + C(t,.)
( 1)
channel, is defined as
Events Beginning Of Train BOT(.) and End Of Train EOT(.), representing beginning and end of a train on the channel, occur when TRAIN (t,*) makes transitions from 0 to 1 and 1 to 0 respectively. Events EOC(OUT1), EOC(OUT2), and EOT(IN) are known as synchronizing events and are used for possible transmissions by a station. A station is said to be alive if it is monitoring synchronizing events successfully and is participating in the process of their assured presence. A station which is not alive is said to be dead. An alive station with a DU to transmit is known as backlogged. It is termed as idle otherwise. The network is said to be alive if at least one of the stations is alive. The network which is not alive is said to be dead. The access protocol executed at a station is as explained below. A station monitors simultaneously OUT1, OUT2 and IN channels. If the carder activity is in progress on OUTI channel, the station waits till EOC(OUT1) occurs. At this instant, if backlogged, it starts its DU transmission and simultaneously monitors activity on OUT1 for possible transmission from upstream. If R
R
Inbound channel (in)
R
D
/'2 S'~
--
SI TI1
R¸
Outbound channel 2 (out 2) /2 $2
72 S~
S
(oat l) o
.
°
.
•
•
°
o
o
•
°
•
S •
•
( a ) Stations in a line
( b ) station I and M ¢olocnted
Fig. 1. Architectureof M-net. Og~ 24-4-B
_
siTII) lf /lll
320
M. Khalid et al.
such a transmission is detected, within td seconds of the transmission, it aborts its own transmission in favour of the upstream one and waits for next EOC(OUTI). It then repeats at every such EOC(OUT1) the above steps. If no transmission from upstream is detected in first td seconds, the station continues its DU transmission. The station thus completes its transmission successfully on OUT1 appending its DU to the end of the partial train. After the transmission of its DU in the train, the station does not sense EOC(OUT1) in the current round and hence will not be able to transmit another DU on OUT1. If the backlogged station observes that the carrier activity is not in progress on OUTI but is progressing on OUT2, it waits for EOC(OUT2). It then attempts transmission of the DU on OUT2. The procedure followed for transmission on OUT2 is the same as that for OUT1. If the station does not observe carrier activities on either of the outbound channels then the station waits till the occurrence of EOT(IN). At this instant, it transmits a special signal for td seconds on OUT1, called locomotive [4], to ensure existence of synchronizing events. It then attempts, if backlogged, access on OUT1. Note that in M-net the end-to-end propagation delay on each channel is same and taps S1 and R of any two stations are connected to the channels in such a way that propagation delay between their S 1 taps is same as that between their R taps. Therefore, the event EOT(IN), used by all stations as the synchronizing event to start a new train, reaches any station exactly at the same time as the beginning of the new train on OUT1 channel because of a station which is upstream on OUTI. Hence, locomotives of all stations overlap on the channel. Following the locomotive transmission, the most upstream backlogged station on OUT1 completes its transmission successfully and the normal MAC procedure follows. The complete train formed consequently is seen on IN by all the stations. If ~'is end-to-end propagation delay on a channel then the propagation delay between S1 and R taps at any given station is 2r. Therefore, the inter-train gap, defined as the time between the end of the last DU in a train on IN channel and the beginning of the next train on it, is (27 +2td) seconds. Thus, stations with indices 1 - m -< M get access opportunities in the order 1 to M on OUTI and then in the order M to 1 on OUT2 in every train. A train, as seen on IN, followed by the inter-train gap constitutes a cycle. Figure 2 shows typical sensing and transmission activities on OUT1 and OUT2 channels and receiving activities on IN channel at a station of index i3. It is assumed that transmission time of a DU, T, is constant. Stations i I and i2 are upstream and station/4 is downstream with reference to station i3 on OUT1. The diagram shows a typical situation when all these stations are backlogged before the end of the partial train and hence transmit a DU on OUT1 as well as on OUT2 in a train. Note that since sensing tap Sm ($2) is upstream the transmit tap T~ (Te) on OUT1 (OUT2), the transmission of i 3 o n OUT1 (OUT2) is not sensed by S~ ($2). Figure 3 shows the corresponding space-time diagram indicating activities on OUT1, OUT2 and IN channels. The EOC(IN) occurs t# seconds after the actual end of transmission on IN channel at a station. The locomotive of the next train is transmitted by the station on the occurrence of EOT(IN) i.e. if no further activity is sensed in next td seconds after EOC(IN). Therefore, the time interval between the actual end of transmission on IN channel at the R tap of station and the start of locomotive on OUT1 by it is 2td seconds. In an alive network, the presence of a synchronizing event is always detected on IN channel in a time interval of (2~" +2td) seconds. If a station (after it becomes alive) does not detect it for (21" +2t#) seconds on IN, it concludes that the network is dead. It then initiates a process of regenerating synchronizing events through a cold-start procedure. This procedure is similar to the one which is followed in Expressnet [4]. The state diagram for the M-net access protocol at a station is shown in Fig. 4. Initially, a station is in dead state. When the request for making it alive is generated, the station starts monitoring carrier activities on all the three channels. If the carrier activity is detected within (2~" +2td) seconds then it is confirmed that the network is alive. In such a case, station follows the normal procedure of DU transmissions and generation of synchronizing events. If no activity is detected on any one of the channels for (27" +2td) seconds then the station confirms that the network is dead. In such a case, it executes cold-start procedure on OUT1 to generate synchronizing event EOT(IN). The presence of a DU at an alive station is indicated by a variable DU_buffer which takes values full or empty depending on the presence or absence of the DU for transmission at the station. As the attempt and defer mechanism on OUTI and OUT2 channel is same, a variable called OUT_CH is defined that takes values OUT1 or OUT2 depending on the outbound channel on which DU transmission may be attempted by the station. If the station, on becoming alive, detects that the network is alive, then it waits for a synchronizing event. At that instant, it moves to appropriate state. However, if the station detects its presence on
Analysis of an asymmetric polling system
321
outbound as well as on inbound channels, it synchronizes with the synchronizing event of an outbound channel. If the carder activity is detected on both the outbound channels, then it synchronizes with EOC(OUT1) for possible DU transmission or for moving to appropriate state. When the station becomes alive, by executing the cold-start procedure, it waits for EOT(IN). At this instant it transmits the locomotive to ensure presence of EOT(IN), even if no station has a DU for transmission in the current train and goes to ready state. The station then attempts (and defers if necessary) the DU transmission on OUT1 if the state of DU_buffer is full at the instant of its moving to ready state. Otherwise, it goes in defer & wait state and returns to ready state if its DU_buffer is full at EOC(OUT_CH). If OUT_CH is OUT1 then the station attempts for DU transmission on O U T I when its DU_buffer is full and it is in ready state. If no transmission from upstream is detected on the outbound channel, then it completes its transmission, sets OUT_CH to OUT2 and moves to defer & wait state. Otherwise, it aborts the transmission, moves to defer & wait state and waits for next EOC(OUT1). If the station does not have a DU till the train has passed on OUT1, and if OUT_CH=OUT1, then the station changes OUT_CH to OUT2. It then waits for EOC(OUT2) and undertakes possible DU transmission on OUT2 following the same procedure as that for the transmission on OUT1. On detection of EOT(OUT2) or after the completion o f DU transmission on OUT2, the station moves to wait state for the start of the next train and the procedure is repeated till the station is alive.
iI
C
12
(t, O U T 1)1 0
2 Transmit 1 tap on out I
iI
Time
EOC (OUT 1) BOC (OUT 1) EOC (OUT l) i2
i3
•
t'~)d~
.
.
Locomotive -"4' k -
Time
C (t, out 2) j . ~ E O C (OUT2)
Prop delay between $I and $2 +ta
Time
i
Transmit tap on out 2
C(t.
P
Time
in) ,
q
~--
Time
I
fl Locomotive Train (t, in)
II
i~
Cycle
--I • ~---2~ + ta - - ~
Time BOT (in) EOT (in) Fig. 2. Activity profiles on OUTI. OUT2 and IN channels with respect to station i3 when stations i, i2, i3 and /4 are backloggedat both the polling instants in a cycle under consideration.
J,
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M. Khalid et al. 3. M O D E L D E V E L O P M E N T OF M - N E T
As explained in Section 2, a server polls stations first in order 1 to M on OUT1 and then in the order M to 1 on OUT2 in a polling cycleJ It serves the station for time T, if it has a DU present in its buffer at its polling instant, before moving to the next station. There are, therefore, 2M polling instants in a polling cycle. Thus, it is modelled as a reverse round robin polling system with the following assumptions: (1) The system consists of M single buffer stations, each of which can hold a fixed length DU. The transmission time of a DU is T seconds. (2) The propagation delay, ~',,, between station m and (m+ l) (1 < - m -< ( M - 1)) is constant. (3) DUs arrive at a station m ( l -< m -< M) with Poisson arrival rate Amand arrival of a DU at a station is independent of arrivals at other stations. (4) DUs arriving at an occupied buffer (backlogged station) or during the service (transmission) of a DU are lost (blocked). A station at its polling instant is either backlogged (station state l) or idle (station state 0). In order to study the performance of M-net, system state is defined as a set of station states at their respective polling instants in a polling cycle. Since the state of a station depends only on its inter-visit time, the state of the system exhibits Markovian property. M-net, therefore, may be modeled as a discrete time Markov chain with a state space of size 2 TM. However, if respective propagation delays between two transmit taps of station M on OUT1 and OUT2 and between the two taps of station 1 on OUT2 and IN are negligibly small, stations 1 and M will have utmost one DU per cycle in steady state. But, station 1 may have DUs when server polls it on OUT1 and OUT2 in the first cycle. Therefore, maximum number of DUs in the first cycle is 2 ( M - 1) + 1 and is L = 2 ( M - 1) in subsequent cycles. Hence, there are 2 L recurrent states.
c~ i2 i3 i4
i4
i3 O
i2
iI
il .= ~2 i3 i4
Time Fig. 3. Space-time diagram of M-net corresponding to Fig. 2. i A polling cycle is defined as the inter-visit time to the same point in the configuration. The convention of starting a cycle from polling instant of station 1 is followed here for convenience.
Analysis of an asymmetric polling system
323
Therefore, the size of the state space, S, of such a system under steady state is 2 L and the dimension of the single step transition probability matrix, P, is 2L× 2 L. ThUS, State of the system may be represented by an L-tuple (sl, s2..... s/) of (0,1) elements. An element in the L-tuple corresponds to the state of a particular station when the server polls it on a particular outbound channel. Let the starting point of the cycle be polling instant of station 1 on OUT2. The cycle, in this case starts when station 1 is polled on OLrI'2 and ends when station 1 is revisited on OUT2. Then, the first element of the L-tuple, sz, corresponds to state of station 1 at its polling instant on OUT2. The element si at i 'h Rosition, for 2--- i -< M, of the tuple indicates state of the i 'h station when the server polls it on OUT1. For (M +1) --< i --<2(M- 1), si at the i 'h position in the tuple indicates state of the ( 2 M - i) 'h station when polled on OUT2. 3.1. Composition of transition probability matrix Let I and J be the system states corresponding to two consecutive polling cycles respectively. An element P(I,J) of single step transition probability matrix P is a product of L terms. Each term, Pi(l,J), 1-< i <- L, depends only on the inter-visit time at the corresponding station. Inter-visit time at a station is a sum of two terms. First term is the walk time of the server between the
C Station dead
'~
l (oo'2:2'.t°b2saJj=iea on all the channels)
Network dead
(
Station alive
(execute cold-s~n
procedure
on out 1)
Network alive
Monitor C (t, out 1 ) = 0 & C (t, out 2) - 0
I~
CI
q
EOT (in)
~ E O C ~ ( o u1), t (out--CH~
=f
Ready
(transmit
I
locomotive, out = CH: - out 1)
[
Train (t, out 1) = 0 & EOC (out 2) (out-CH: = out 2)
DU buffer = empty -
)
~
(o
] DU_buffer : f u l ~ ~ y [ (start DU TX. on "~totf.
(t-t~,out_OH) - 0 (complete D U TX.)
[ EOC (out_CH) [ DU_buffer = ] empty
C
out_CH - out I &
train (t, out 1) - 0 out_CH = out I out 2)
(out_CH: out_OH = out 2
(Transmit) EOT (out 2) Fig. 4. State diagram of M-net access scheme.
(out-CH: = out 2)
M. Khalid et al.
324
consecutive visits to the station. The second term corresponds to the transmission time incurred in serving the DUs at the backlogged stations during the consecutive visits. Walk-time o f the server between the M-I
consecutive visits to rn th station (1<- m <-- ( M - 1)) on OUT1 first and then on OUT2 is '6m,M)=2 E
k=m
r~. Similarly, walk-time o f the server between visits to station m (2 < - m <- M) on OUT2 and the next visit m-I
to the same station on OUT1 is "6m,1)=2 £
r k. Walk-time for end stations is 2r. The number o f DUs,
k=l
K, served during the inter-visit time at a station corresponding to i 'h polling instant, ri(K), depends on the system states I and J corresponding to two consecutive polling cycles. The inter-visit time ~(K) is, therefore, given by
2r+KT
for i= 1 and 0-
"6i,1)+KT
2(M- lj j=2
sfl)<--2M- 3
for 2<-i<--Mand 2 ( M - 1)
~(/0 =
£
0
i- 1
sfl)+ £ sj(J)<--2i- 3
j = ( 2 M - i+ 1) -
(2)
j=l
"62M - i,M) + KT for M+ 1<--i.~2(M- 1) and 0
~
s~(J)--<2(i- M) - 1
j=t2M-i+l)
-
As can be seen from equation (2), r/(K) for a given value o f K is station index dependent. Moreover, the maximum value o f K is also a function o f i and M. Thus, the i 'h term o f P(I,J), P~(I,J), is given by
Pi(l,j)= {
e-a'~'~x)
if si(J)= 0
1 - e - a'~
(3)
Note that for (M + l) -< i < - 2 ( M - 1), Ai=AEU_i, since each station is polled twice in a cycle by the server. Therefore, an element P(I,J) can be written as 2 ( M - I)
P(I,J)=
I1 [e-a,',~x).6(si(J))+(1-e-a,',~x~).si(J)]
(4)
i=l
Where S(.) is a Dirac delta function.
3.2. Steady state probabilities Let p(d) be the steady state probability of a system state and p be the associated steady state probability vector o f size 2 L. The system state probabilities can be determined from
p [ l - P] =0
(5)
where I is the identity matrix and 0 is a null vector o f dimensions (2 L × 2 L) and 2 L respectively.The necessary and sufficient condition for equation (5) to have a non-trivial solution is that rank of [I - P] should be less than 2 L [15]. It is shown in Appendix A that the rank o f [ I - P ] i f ( 2 L - 1). As the rank o f [I - P] is exactly one less than the number o f elements in p, solution o f equation (5) will give ratios o f steady state probabilities. Applying normalization relationship on sum o f the probabilities, steady state probabilities can be evaluated from these ratios. The procedure for finding steady state probabilities thus consists o f performing linear operations on rows o f [I - P] to get an upper triangular matrix. If these operations do not change the value o f the determinant o f the resultant matrix, then the errors due to numerical computations are reduced. 2 It may be noted that for M= 2 and a~ = As = A, M-net with stations at the two ends o f OUT 1 degenerates to corresponding Expressnet. Its state probabilities as adopted from [5] with M = 2 are: X4
Pt°'°)= 1 - x2 +x2y
(6)
2One way to achieve this is by using Gauss elimination technique. An alternate way to evaluate an element of the upper triangular matrix directly from [! - P] is to take ratios of the values of two determinants of the appropriate matrices of dimensions (1,/) and (1- I, 1- 1) [16]. However,this approach, in general, takes more computational time.
Analysis of an asymmetric polling system
325
p~o.l)=p.,o)=(x -2_ l).p~0.0)
(7)
p..i)=p
(8)
where x = e ~ and y = e at.It can be shown that equations (6)-(8), satisfyequation (5) verifying the validity of the analysis approach, proposed above, for M=2. 4. P E R F O R M A N C E
MEASURES
OF M-NET
Major performance measures of a network are channel utilization,mean and variance of delay, and the blocking probability. In the following subsections methods for calculating these measures are explained. 4.1. M e a n channel utilization Let pN(n) be the probability mass function (pmf) of the number of DUs, N, served in a polling cycle. Then pN(n) is given by L
p~(n)= Y~ p(J).8( ~ s , ( J ) - n) J~S
(9)
i= I
The average number of DUs served in a cycle, E(N), is given by 2 ( M - I)
E(N)=
Y~ np~(n)
(10)
n=0
The train on inbound channel is followed by an inter-train gap of (2~" +2ta) seconds. Normally, td is a very small fraction of T and hence can be neglected in calculating the channel utilization. Therefore, the mean channel utilization, CM, of the network is given by E(N) T Cu= E(N)T+2r -
1 2a I+--
(11)
E(N)
where t~= r/T. Using this relationship, mean channel utilization profile of M-net under different values of a and arrival rates can be obtained. 4.2. Mean and variance o f delay at a station As every station is polled twice in a cycle, DUs arriving at a station may be served on either of the outbound channels. All intermediate stations may transmit none, one or two DUs in a cycle. However, end stations will have utmost one DU per cycle because the time latency between the server's departure instant from OUT2 (OUTI) and the subsequent polling instant on OUT1 (OUT2) for station 1 (M) is assumed to be negligibly small. A DU at a station experiences delay because of channel access waiting time and its transmission time on it. Mean and variance of the delay in M-net, even for identical arrival rates at all stations, are station index dependent because the respective inter-station walk times from OUT1 to OUT2 and from OUT2 to OUT1 in a polling cycle are station index dependent. The probability, p(si= 1, K), that K number of DUs will be served during the inter-visit time ~,.(K) and si= 1 is E W
2(Mp(Jll)p(l)8( j=2
1)S)(1)
- IO
YJ~,(J)= 1
f o r i= 1 and O<--K<--2M- 3 p( Jll)p( l) 3(X j.2u _ <,_ , ~s,(I ) + X j,, , s,(J) - l O W
¥,/t*j(,/)= 1
p(si= 1,K)=. for 2<-i<-M and O<-K<-2i - 3
x
~¢J1$~1)= I
(12)
p(J38(X~2~,-.-l:j(J)-g)
f o r (M+ 1)--
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M. Khalid et al. 2M - 3
~, p(si=l,h')
fori=l
K=0 2i-3
~, p(s~= 1,K) p(si= 1)=' x.o
for 2<-i<-M
(13)
2(i- hi0- 1
E
p(s~=l,K) for M+ l < i < 2 ( M - 1)
K=0
and the conditional probability, p(Klsi = 1), that K number of DUs are served during the station inter-visit time that has elapsed prior to i 'h polling instant, given that the corresponding station is backlogged at its polling instant, is p(Klsi= 1)= p(si= l, . p(si= 1)
1<-i<-2(M - 1)
(14)
Where K is a non-negative integer such that (2M- 3 for i=1 K<~2i- 3 for 2<--i<--M 12(i - M) - 1 for (M + 1)--
(15)
The probability density function, fw,(wlK), of the access waiting time W~of a DU served at i 'h polling instant for a given r,(K) is [17] A,e ~'w fw~(wIK) = ~ e A ~ 1 0
O<--w< ~i(K)
(16)
otherwise
Where ~,(K) is given by equation (2). The conditional mean and second moment of W,, for 1<- i -<2(M- 1), are then given by E(WilK)= 7/(K)ea'~'
1 hi
(17)
and
l
E(W21K)= e ~ ' ' ~ - 1
[ r2(K)ea,T,(to 2~(K)ea'"<~ 2e~'"¢~° ,~22 ] ,~i + A~
(18)
Note that as A,.---, 0% right hand sides of equations (17) and (18) reduce to ~(/Q and r2(K), the inter-visit time and square of the inter-visit time at i ,h polling instant respectively, and the delay expression matches with the bounds derived in [1]. Similarly, as At---* 0, equations (17) and (18) reduce to ~ g ~ and ~'Ttt~) 3 I respectively and equation (16) reduces to a uniform probability density function ~ for 0 <- w - %(K) and is zero otherwise. The mean and second moment of waiting time, E(W~) and E(W~i), for ( 1 - i -<2(M- 1)) are given by E(W,)= E E(W, IK)p(KIs,= 1)
(19)
VK
and E(W2)= ~ E(W~IK)p(KIs~= 1)
(20)
VK
Where K is a non-negative integer satisfying equation (15) Let to~.,, and to~.mbe the waiting time at a station m, when a DU is served on OUT1 and OUT2 respectively. Then, their mean values are given by E( toij,)= E(Wm) for 2<-m<-M and
(21)
Analysis of an asymmetric polling system
327
f E(W.~) for m = 1 E(oh~,,)= / E(W~u_.) for 2<-m<-(M- 1)
(22)
A DU may be served on OUT1 or OUT2. Therefore, the mean, E(to.), of the waiting time, to=, at a station m (1-- m --< M), is expressed as
E( o~.,,,) E( w,,,)=
for m = 1
p(s,. = 1) p(s2M-.= 1) p(s,,,= 1)+p(sz~_,.= 1) E(w.j.)+ p(sm= 1)+p(s2M-,.= 1) E(o~.) for 2<--m<--(M- 1) (23)
for m =M
E(toj.m)
Similarly, expressions for second moments of the waiting times. E(tozt.,.).E(w~.m) and E(to~.), can be deduced. Let D~.m and D2.,. be the normalized delay at a station m, normalized with respect to DU transmission time T. when a DU is served on OUT1 and OUT2 respectively. Then, the mean of Dt..~ and D2.mare given by (24)
E(Dl.m)= T [E(wLm)+T]for 2<-m<-M and 1
E(D2~.)= ~ [E(toz~)+ T] for l <-m<-M - 1
(25)
The mean, E(Dm), of the normalized delay at a station m, D,., (1~ m --< Mr) is
E(Dz.m) E(D,.) =
for m= 1
p ( s m = 1) p(S2M-,~= 1) p(sm = I ) +p(s2u-.,= 1) E(D,~)+ p(sm= 1)+p(s2M-~,= 1) E(D2~) for 2<--m<--(M- 1) (26)
E(Dw.)
for m=M
The variance of the normalized delay at a station m, o~o~,(1-< m -< M) can then be easily expressed as 2 _ O"w.2 ' _-O'Dm--
~ I [E(a,m2 ) _ E2(wm)]
(27)
4.3. Blocking probability at a station When a DU arrives at a station, its buffer gets full. DUs generated during the time interval in which the buffer at a station is not empty are lost (blocked). Therefore, the blocking probability (or probability of loss) at a station is another important performance measure in a single buffer system. The blocking probability at a station served at the i ~ polling instant is [17] pb(i)=
XI(E(W,)+D 1 + X~(E(W3+ r )
for 1-
(28)
A DU stored in the buffer may be served on either of the outbound channels. Therefore, the blocking Table I. End-to-end distance and DU size for various values of a ot
End*to-end distance (Kin)
Number of bits in a DU
0. I 1.0 I0 100
0.2 2.0 20.0 20.0
1000 1000 I000 100
328
M. Khalid et al.
No. of stations (/14)= 5 end-to--end prop. delay
a 0.80 -
/
/
packet transmission time •
a=O.l
/ f
/
t
/ f a = l
M-net -- Expressnet
O
.,.
e"
~
o.40-
~
1
a=10
a= 100 0.00
I
I0
I
I
[
,
I
I O0
1000
10,000
100,000
Lambda (arrivals/s) Fig. 5. Mean channel utilization versus arrival rate for M-net and Expressnet with M=5 and or=0.1, I, 10 and 100.
probability at a station with station index m, pB(m), is given by Am(E( W~) + T)
f o r m= l and M
I + A~(E(W.~)+ T)
pB(m) =
A~(E(WI)+ T)
I + a,.(E(Wm)+ T) +
×
(29)
p(Sm = 1)
p(Sm= 1) +p(szAc_m= 1)
Am(E(W~_ m)+ T) p ( s ~ _ m= 1) × 1 + Am(E(W2M_m)+T) p(sm= 1)+p(Sm-m= 1)
for 2<--m<--M- 1
5. NUMERICAL COMPUTATIONS AND THEIR INTERPRETATION The analysis approach presented in Section 3 and 4 is illustrated here. The performance of M-net and Expressnet (as a representative example of round robin system) with equally spaced stations and identical arrival rates, A, are evaluated for this purpose. Results are obtained for four different values of a i.e. a=0.1, 1, 10 and 100. Table 1 shows end-to-end distance on one of the channels and number of bits in a DU for various values of a. The propagation delay and transmission rate are assumed to be 5 microseconds/Km and 100 Mbps respectively. DU size is 1000 bits for a=0.1, 1 and 10. For a = 100, however, DU size is 100 bits. Alternatively, one could have assumed a constant DU size of 1000 bits for all values of a. However, for a transmission rate of 100 Mbps, this would mean end-to-end distance of 200 Kin at or= 100. Figure 5 shows variation of mean channel utilization versus Poisson arrival rate A with a as a parameter for a system of M= 5 stations. At lower values of A, difference in channel utilization of M-net and Expressnet is insignificant. For example, the performance of the two systems is almost same for values of A up to 1000. However, as A increases, M-net has better channel utilization than the corresponding Expressnet system. This is because in M-net, at lower values of A, the probability with which a station transmits on both channels in a cycle is very small. However, as A increases, the
Analysis of an asymmetric polling system
329
probability with which a station utilizes both access opportunities in a cycle increases. Finally, at very large values of A, the channel utilization, CM, of M-net and, Ce, of Expressnet saturate to values corresponding to the situation in which stations are backlogged at all the polling instants in most of the cycles. Thus, as A----, 0% CM---* Cu(max) and C e ---* Ce(max) where CM(max)=l/[1 + a / ( M - 1 ) ] and Ce(max) = 1/[ 1 + 2a/M] are the bounds on channel utilization in M-net and Expressnet respectively Ill. Figure 6 shows the variation of normalized mean delay at stations with A for a system of M= 5 stations and a = 10. Delay performance at all stations in symmetric Expressnet is same. However, mean delay at a station, in M-net, is station index dependent. In M-net, delay performance E(DI)=E(D2,1)at station 1 and E(Ds)=E(Dj.5) at station 5 are same. At lower values of A, mean delay at end stations in M-net are same as that for Expressnet. This is because, at these values of A, the inter-visit time at end stations in M-net and Expressnet are comparable. However, at higher values of A, inter-visit times for these stations in M-net are more than that in Expressnet. Therefore, delays experienced at these stations are also more than that in Expressnet. Delays at intermediate stations in M-net are smaller than the delay in Expressnet for all values of A. It can be seen that at station 2, for a given A, mean delay on OUT1, E(DI.2), is less than mean delay on OUT2, E(O2.2). This is because ~-(2,1) < 7-(2,5). For station 3, delays on both outbound channels, E(Dj.3) and E(D2,3), are same and are equal to the mean delay, E(D3). Mean delay at station 3 is less than the mean delays at stations 2 and 4 over a large range of h. It is approximately half of the Expressnet mean delay. This is because walk time to the central station in M-net is half (or close to half if M is an even number) as compared to that in Expressnet. As the arrival rate increases, the intervisit time at a station also increases resulting in increased delay. As A ---. o% mean delays at all stations in M-net, except end stations, are same. This happens because at such high values of A, as soon as the server departs a station, DU arrives at the station with very high probability. In such a case, mean delay at a station approaches the inter-visit time and a DU will be served on either of the outbound channels with equal probability. This observation is also in agreement with the results derived for bounds on 30-No. of s t a t i o n s (M) ffi 5
E (DI), E (D 5)
alpha = 10 f
M-net 25 --
i
_ Expressnet
/
/
""
"-
~
/
/ /
~
~/
E (D2,2), E (/)1,4)
20 ell
it
I
o
15 o N °~ o z 10 E (DI,2), E (Di,4)
5--
0
,
l0
J
i i ,,,,I
,
I00
i
, , Jllll
,
I000
i
i ~ iiitl
I
10,O00
,
, It,Hi
100,000
,
,
, , ,,,,I
i
1,000,O00
Lambda (arrivals/s) Fig. 6. Normalized mean delay on O U T I and O U T 2 at stations and normalized mean delay at stations versus arrival rate for M-net and Exprcssnet with M = 5 and a = 10.
M. Khalid et al.
330
maximum normalized delay in [ 1]. The maximum values of mean delays, on individual channels at intermediate stations, in M-net and mean delay at any station in M-net and Expressnet in Fig. 6, are in exact agreement with the bounds derived in [1]. As per [1], maximum normalized delays at an intermediate station m, on OUT1 and OUT2, ale Dj~(max)= [1(re, l) + 2 ( m - 1)T]/T, and D~(max)= [1(re,M) +2(M-m)T]/Trespectively. The maximum value of mean delay at an intermediate station in M-net is Dm(max)=(a + M - 1). For end stations in M-net, it is 2(a + M - 1) and is (2a + M) for a station in Expressnet. Variation of normalized mean delay with mean channel utilization for a = l0 with M and station index as parameters is shown in Fig. 7. Variations for M=5 arc abstracted from Figs 5 and 6. Similarly, respective profiles for M=3 and 4 are also depicted in the figure. For the same channel utilization, delay performance of M-net is better than that of Expressnet, even at end stations. This is because, for the same channel utilization in both the cases, the arrival rate in Expressnet is higher than that in M-net (see Fig. 5). This results in lower mean inter-DU arrival time for Expressnet than in M-net. In M-net, delay at end stations for M=5 is lower than that for M=3 and 4 except for lower values of channel utilization. This is because for the same channel utilization, A for M=5 is smaller than that for M=3 and 4. Thus, in such a case, the mean DU inter-arrival time for M=5 is greater than that for M=3 and 4. Figure 8 shows the variance of delay versus mean channel utilization for a system of M=5 stations and a = 1 and 10. In M-net, for a given value of a and channel utilization, variance is highest at end stations and lowest at the middle station. This is because of the reduction in the number of stations that the server visits during the inter-visit time for middle stations as compared to the end stations. These stations may or may not have DU at their respective polling instants. At lower values of channel utilization, the variance of delay at a station is non-zero because of randomness between the arrival instant of the DU and the instant at which server polls the station. The variance at a station drops to zero at saturation (i.e. as A --* oo), since a new DU arrives as soon as the server departs the station, transmitting the previous 30 - No. o f s t a t i o n s ( M ) = 3 , 4 , 5 alpha = 10 25 --
m--
Mr5 m=l,5
M-net Expressne!
M=4
'/
M=3]
I
I
m = 1,3
I I
I I I
20
m = 1,4
?/ /
o
.
15 M
es O
ffi4
IM ffi5
M=3 m=2
Z
I0
01 0
I
I
I
I
I
I
0.05
0.10
0.15
0.20
0.25
0.30
Mean channel utilization Fig. 7, N o r m a l i z e d m e a n delay versus m e a n channel utilization for M-net and Expressnet with a = 10 a n d Mffi3,4 a n d 5.
Analysis of an asymmetric polling system
331
DU, and all cycles are of maximum length. Variation in blocking probability at a station with arrival rate A is shown in Fig. 9 for M = 5 and a = 1 and 10. At lower values of ~, the blocking probability at all stations is negligibly small. As the value of A increases, the blocking probability also increases. This increase is higher for a = 10 than for a = 1 caused by a higher cycle time for a = 10. Over a large range of A, blocking probability at end stations in M-net is larger than that in Expressnet. However, at intermediate stations in M-net it is smaller than that in Expressnet. This is as a consequence of delay performances at these stations. 6. CONCLUSIONS
In this article, a model of a reverse round robin polling system is developed for the analysis of a UBS LAN called M-net. It is modeled as a discrete time Markov chain with a state space of size 2 2tM-t) for a system of M stations on the network. It is assumed that stations have a single buffer with Poisson arrivals. It is found that the system is inherently asymmetric in nature because the inter-visit time at a station is a function of station index m. State transition probability matrix approach is used to calculate system state probabilities through the solution of 2 2tM-m)linear simultaneous equations. Expressions for mean channel utilization, mean and variance of delay and blocking probability at a station are derived in terms of these state probabilities. The analysis approach developed in this article can be used to evaluate round robin as well as reverse round robin systems in a unified manner. Finally, the performance of M-net for a system of 3, 4 and 5 stations is numerically evaluated and compared with those of Expressnet as an illustration. Expressnet is taken as a representative example of LANs based on round robin system. It is found that the performance of M-net is better than that of Expressnet at higher values of A and is comparable otherwise. The queuing analysis presented in the article is applicable to a class of engineering systems that can be modelled as a polling system with single buffers and round robin or reverse round robin scheduling. 5O
No. of stations (M) = 5 alpha = 1.0, 10 40
,;Y
2" 30
>
2O
M-net - - -- Expressnet
!
', I I I I I I I
10
I I
0.0
0.2
0.4
0.6
0.8
1.0
Mean channel utilization Fig. 8. N o r m a l i z e d variance o f delay versus m e a n channel utilization for M-net a n d Exprcssnet with a = i, 10
and M=5.
332
M. Khalid et al. 1.0 No. of stations (M) = 5 alpha = 1.0. 10 M-net -- Expressnet 0.8
0.6 es a e~
.o
o
0.4
= i.5 = 2,4 =3 =1
a=lO,
m-- 1,5-
0.2 m
~.
01=
10
100
1000
I 0,000
100,000
1,000,000
Lambda (arrivals/s) Fig. 9. Blocking probability versus arrival rate for M-net and Expressnet with M=5 and a = 1 and 10.
REFERENCES 1. Khalid, M., Vyavahare, P. D. and Kekre, H. B., M-net: Architecture and access protocol. Technical Report TR-132-94, Dept. of Computer Science and Eng., Indian Institute of Technology, Bombay 400 076 INDIA, February 1994. 2. Levy, H. and Sidi, M., Polling systems: applications modelling and optimization. IEEE Trans. Commun. , 1990, 38, 10 1750-1760. 3. Takagi, H., Applications of polling models to computer networks. Comp. Networks and ISDN Sys., 1991, 22, 193-211. 4. Tobagi, E A., Borgonovo, E and Fratta, L., Expressnet-a high performance integrated service local area network. IEEE J. Select. Areas Commu~. 1983, SAC-l, 5 898-913. 5. Fine, M. and Tobagi, E A., Performance of unidirectional broadcast local area networks: Expressnet and Fasnet. IEEE J. Select. Areas Comraun., 1983, SAC-I, 5 913-925. 6. Gerla, M., Rodrigues, P. and Yeh, C., Buzznet: A hybrid random access/virtual token local network. Proc. of the Globecom, 1983, December, 1509-1513. 7. Rodrigues, P., Fratta, L. and Gerla, M., Tokenless protocols for fiber optic local area networks. IEEE J. Select. Areas Commun., 1985, SAC-& 6 928-940. 8. Abeysundara, B. W. and Kamal, A. E., High speed local area networks and their performance--a survey. ACM Computing Survey, 1991, 23, 2 221-264. 9. Economou, E. G., Bolls, S. and Philokyprou, G., FBC-net: a reverse round robin LAN for voice and data traffic. Comp. Commun., 1991, 14, I 44-52. 10. Economou, E. G. and Bolls, S., G-net: Implicit token-passing bus LAN. Electronics letts, 1991, 27, 16 1449-1451. 11. Vyavahare, P. D., Khalid, M. and Kekre, H. B., Recent trends in implicit token based local area networks. Proc. IEEE Int. Conf. on Computer Networks, ICCN-92, 1992, December, FT30.-40. 12. Mack, C., Murphy, T. and Webb, N. L., The efficiency of N machines unidirectionally patrolled by one operative when walking time and repair times are constant. J. Roy. Star. Soc., 1957, 19B, 166-172. 13. Takagi, H., Queueing analysis of polling systems. ACM Computing Surveys, 1988, 20, 5-28. 14. Takine, T., Takahashi, Y. and Hasegawa, T., Exact analysis of asymmetric polling systems with single buffers. IEEE Trans. Commun., 1988, COM-36, 10 1119--1i 26. 15. Bellman, R., Introduction to matrix analysis. McGraw Hill, 1960. 16. Grotschel, M., Lovasz, L. and Schrijver, A. Geometric algorithms and combinatorial optimization. Springer-Verlag, Berlin, 1980. 17. Kaye, A. R., Analysis of a distributed control loop for data transmission. Proc. Syrup. Comput. Commun. Networks TeletrqO~c, 1972, April, 47-58.
A n a l y s i s o f an a s y m m e t r i c polling s y s t e m
333
APPENDIX A THE RANK OFA STATE TRANSITION MATRIX Theorem 1. Let the single-step transition probability matrix of a L-state irreducible, aperiodic Markov chain be P. Then rank of matrix [ l - P ] is ( L - l). Proof The chain is aperiodic and irreducible, therefore O
(30)
Where P(i,j) is the (i,j)'* element of the transition probability matrix P. Then, the sum of all columns of[P - I] is a null column vector. Therefore, L columns of matrix [P - I] are linearly dependent. Hence, the rank of [P - I] must be (L - 1) or less. Let fl~, ~ ..... tL be scalars, not all of which are zero, such that j=l
t~P(ij) - ti=O f o r all !
(31)
Let t , be a scalar among t l to tL which satisfies the following property likl>-Itjl f o r all i
(32)
Without loss of generality, one can assume that ilk= 1. Consider k th row of the matrix [P - I] which is given by [e(k, 1),P(k,2),...P(k,k) - l,...e(k,L)]
(33)
( t j P ( k , i )+ /$zP(k,2)+ ... + t~P(k,k)+... + tLP(k,L) ) - ik=O
(34)
From equation (3 i) one gets
As t , = 1; ~ P(k,j)= 1; O<]P(i,j)< 1 f o r all 1 <~i,j <- L and t i < 1 f o r all 1< i < L, therefore there is only one solution to equation j=l
(34) given by fl, = ~ . . . . . tk . . . . iL= 1
(35)
Hence k '~ column of matrix [I - P] can be represented by a unique linear combination of the remaining (L - 1) columns. Therefore there are ( L - 1) independent columns in matrix [I - P]. As the column rank of a matrix is same as the row rank, therefore rank of the matrix [I - P] is (L - 1).