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Delay analysis of a worst-case model of the MetaRing MAC protocol with local fairness
COMCOM 1230
Computer Communications 20 (1997) 671–680
Delay analysis of a worst-case model of the MetaRing MAC protocol with local fairness G. Anastasi*, M. La Porta, L. Lenzini University of Pisa, Department of Information Engineering, Via Diotisalvi 2, 56126 Pisa, Italy Received 3 November 1995; accepted 16 February 1996
Abstract The MetaRing is a medium access control (MAC) protocol for gigabit LANs and MANs with cells removed by the destination stations (slot reuse). Slot reuse increases the aggregate throughput beyond the capacity of single links but may cause starvation. In order to prevent this the MetaRing MAC protocol includes a fairness mechanism. Two types of fairness algorithms have been proposed: ‘global’ and ‘local’. The MetaRing analysed in this paper implements the local fairness algorithm specified in [2]. In order to reduce the complexity we have identified a simplified model which can be analytically solved and yet still provides useful information on network performances. This model represents a worst-case scenario in which network congestion is stressed, i.e. no station, apart from a specific station (tagged station), ever has an empty queue. The model proposed can be represented by a discrete time discrete state Markov chain of M/G/1-type and hence the matrix analytical methodology has been used to solve it. Our analysis focuses on the average access delay experienced by the tagged station as a function of the offered load. The results show that average access delay remains bounded for values of offered load less than or equal to 90%. Furthermore, the average access delay depends on the number of interfering stations and on the protocol parameter Q r. q 1997 Elsevier Science B.V. Keywords: MetaRing; Gigabit MANs; Local fairness
1. Introduction The MetaRing is a MAC protocol for high speed LANs and MANs with spatial reuse, which can achieve an aggregate throughput much higher than the network capacity [8]. However, spatial reuse may cause starvation at stations covered by upstream highly loaded stations. A fairness mechanism is thus required to prevent this from occurring. Fairness protocols have already been proposed and they can be classified into ‘global’ [3] and ‘local’ [2,6]. A globalfairness algorithm regulates access to the shared media by considering the network as a single communication resource. By contrast, a local- fairness algorithm views the media as a distributed collection of communication resources. Hence, a local fairness algorithm regulates the transmissions of the interfering stations without affecting the others so that local fairness algorithms can exploit the advantages of spatial reuse better than global algorithms. A simulative analysis of a MetaRing MAC protocol with a local fairness algorithm was performed in [2], where the * Corresponding author. Tel.: +39 50 568511; fax: +39 50 568522; e-mail [email protected]
0140-3664/97/$17.00 q 1997 Elsevier Science B.V. All rights reserved PII S 01 40 - 36 6 4( 9 7) 0 00 5 6- X
network was analysed in asymptotic conditions (i.e. when stations always have traffic to send). Station and aggregate throughputs were measured for various network scenarios. The results reported in [2] show that the throughput achieved by a station is not influenced by the behaviour of stations which do not interfere with it. Thus, stations in the networks can be divided into disjoint groups. Within a single group the throughput achieved by each station, as well as the aggregate throughput (i.e. the sum of the throughputs of all the stations in the group), strictly depends on the interactions among stations belonging to the group. In [1] we focused on a network scenario which is realistic but particularly critical with respect to slot reuse. It consists of a single group of N þ 1 stations located in one half of the ring. The interference between stations is maximized since stations {1} to {N} are assumed to send traffic only to station {N þ 1} which operates as a gateway to an external network (Fig. 1). The analysis was divided into two parts. In the first part, the network was analysed in asymptotic conditions, and closed formulae for the aggregate and station throughputs were derived. In the second part, a thorough simulation analysis of the network was performed in underload (i.e.
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G. Anastasi et al./Computer Communications 20 (1997) 671–680
Fig. 1. Network scenario.
when the offered load is lower than the network capacity) and overload (i.e. when the offered load is greater than the network capacity) conditions. The rationale behind the simulative choice was that the modelling and performance analysis of the MetaRing MAC protocol with local fairness is very difficult. In fact, an exact model of the network should take into consideration interactions among the stations in the network. This makes an analysis almost impossible. To overcome these difficulties in the present paper we identify a simplified model of the network scenario shown in Fig. 1. This model can be analytically solved and yet provides useful information on network performance. This model represents a worst-case scenario in which network congestion is stressed (i.e. no station, apart from a specific station, hereafter referred to as the tagged station, ever has empty queue). We will refer to this model as the worst-case model, which will be analysed using the embedded Markov chain technique. We will show that the transition probability matrix of this embedded Markov chain is of M/G/1-type [7]. We will thus apply the matrix analytic methodology to calculate the stationary probabilities of the embedded Markov chain. This paper is organized as follows. Section 2 outlines the MetaRing MAC protocol and the local fairness algorithm. Section 3 describes the worst-case model. Section 4 is devoted to the solution of the model, while the results obtained are reported and discussed in Section 5. Conclusions are drawn in Section 6.
2. The MetaRing MAC protocol with local fairness The MetaRing is a medium access control (MAC) protocol for LANs and MANs operating at a speed of 1 Gbit/s and above [8]. It connects a set of stations by means of a bi-directional ring made up of full duplex point-to-point serial links. The protocol provides two types of services, asynchronous and synchronous [9]. This paper only deals with the asynchronous type. A MetaRing network can operate under two basic access control modes: buffer insertion for variable size packets, and slotted for
fixed length cells. In this paper only the slotted access mode is considered. According to the slotted access mode, information is segmented into cells and each cell transmitted in one slot. Slots are structured into a header and an information field. The header includes a busy bit which indicates whether the slot is empty (busy bit ¼ 0) or busy. A cell can be accommodated on an empty slot. Since the ring is bi-directional the MetaRing MAC protocol uses the shortest path criterion to choose one of the possible directions. Cells are removed by the destination station which frees the slot by resetting the busy bit to zero. After cells have been removed, the slot can be reused by the same station or by its downstream stations. Slot reuse increases the aggregate throughput beyond the capacity of single links but may cause starvation. To prevent this from occurring a fairness mechanism is required. The fairness algorithm analysed in this paper is local and is triggered by any station whenever it foresees a potential starvation. The algorithm is based on two control signals: REQ (REQuest) and GNT (GraNT) which propagate in the opposite direction with respect to the information flow they regulate. According to this algorithm a station can be in a finite set of states and, correspondingly, in two different operation modes, non-restricted and restricted. In the former there are no limitations on the number of cells a station can transmit, while in the latter a station can only send a predefined quota, Q r, of cells. Note that in [2] the condition under which a station establishes that it will become starved is not specified. Therefore, to carry out our analysis we had to integrate the local fairness algorithm specified in [2] with the following definition: a station observing more than Q s contiguous busy slots becomes starved. Clearly, other definitions are possible and each one may lead to different performances. With our definition of starved station, the local fairness algorithm works as follows. If no conflict occurs (i.e. if no transmitting station is covered by the traffic originated by upstream stations) all the stations are in the free access (FA) state. In this state a station operates in non-restricted mode and therefore it can transmit a cell every time it observes an empty slot. Once a station (say station {i}) becomes starved, it triggers the local fairness mechanism by sending the REQ signal to the upstream station {i ¹ 1} and then enters the tail (T) state. Upon reception of a REQ, station {i ¹ 1} enters the restricted mode of operation and the head (H) state. Then, if station {i ¹ 1} itself observes busy slots from upstream stations it immediately forwards the REQ to station {i ¹ 2} and enters the body (B) state. While in states T, B and H a station can send at most a quota Q r of cells (restricted mode). Upon satisfaction (i.e. transmission of the predefined quota) or as soon as the buffer becomes empty, the tail station sends a GNT signal to its upstream station and switches back to the non-restricted FA state. Upon receiving this GNT, the upstream station follows similar rules: if it is
G. Anastasi et al./Computer Communications 20 (1997) 671–680
in state B, it transits to state T and will forward the GNT upon satisfaction or when its buffer becomes empty (whichever occurs first). If a station is in state H, it switches to state FA and the local fairness cycle involving a set of stations located on that ring segment is terminated. The local fairness mechanism thus creates a request path which contains unique and distinct head and tail stations for each segment in the ring where there are interfering stations. Furthermore, since there might be multiple initiators of the fairness algorithm (and, therefore, of a request path), two or more distinct request paths may overlap. When this occurs the overlapping request paths are merged into a unique request path. Further details on the local fairness algorithm can be found in [2].
3. Worst-case model description To reduce the complexity of the MetaRing MAC protocol modelling, we focus on station {N} (tagged station) and we assume that the remaining stations (i.e. stations {1}, {2}, …, {N ¹ 1}) operate in asymptotic conditions (i.e. they always have cells to transmit). Furthermore, we assume that the distance between stations {j} and {j þ 1} (j ¼ 1, 2, …, N ¹ 1) is constant and equal to an integer number d of slots. Thus, we consider a worst-case model which is analytically tractable and yet provides useful information on network performance. We begin the description of the worst-case model by
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analyzing the slot occupancy pattern observed by the tagged station. In order to make this easier we first derive the slot occupancy pattern by assuming that the tagged station is idle (it never has traffic to send). Then, we investigate how the slot occupancy pattern varies when the tagged station becomes active. 3.1. Slot occupancy pattern observed by the tagged station Under the hypothesis that the tagged station is always idle the network operates in asymptotic conditions and, thus, the considerations reported in [1] can be applied. Specifically, the network behaviour is cyclic and can be represented by means of Fig. 2, which describes the operations performed by each station. Without losing any generality, we can observe the evolution of the network since the instant at which the station {N ¹ 1} forwards the GNT signal to the upstream station entering state FA. It can be shown that at this instant all the upstream stations are satisfied [5]. Hence, when station {j} (j ¼ N ¹ 2, N ¹ 3, …, 2) receives the GNT from a downstream station, it switches from state B to T and, immediately after, sends the GNT. Then, it transits to state FA and enters the non-restricted mode. The behaviour of station {1} when GNT arrives is slightly different from the other stations. Since this station is the head of the request path it immediately transits to state FA. The above considerations imply that, after GNT has been released, station {j} (j ¼ N ¹ 1, N ¹ 2, …, 2) observes a
Fig. 2. Station operations when the tagged station is idle.
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train of 2d empty slots. Since {j} operates in non-restricted mode it uses all of them before being covered by the upstream stations. Now, let us analyse the REQ propagation. When station {N ¹ 1} is covered by the upstream traffic it counts for Q s þ 1 busy slots and then performs the following operations: it sends a REQ upstream, switches to state T and enters the restricted mode of operation. Stations numbered from {N ¹ 2} to {2} all execute the same operations. Each of them receives the REQ from the downstream station and at the same time each observes the (Q s þ 1)-th consecutive busy slot (and hence they would be ready to send a REQ upstream). The simultaneous occurrence of the two events is managed by each of these stations as follows: they first move from state FA to H and, immediately after, forward the REQ upstream entering state B. The trajectory of the REQ is thus parallel to the trajectory of the GNT as shown in Fig. 2. Since station {N ¹ 1} issues a REQ 2d þ Q s þ 1 time slots after GNT release, the REQ itself reaches station {1} exactly 2d þ Q s þ 1 slots after GNT arrival at station {1} itself. This implies that station {1} transmits 2d þ Q s þ 1 cells in state FA before receiving a REQ from the downstream station. Upon reception of the REQ, station {1} changes its state from FA to H, enters the restricted mode, and sends Q r more cells before stopping transmitting because it has been satisfied. After a time interval equal to d slots, station {2} observes empty slots and thus it sends its quota (Q r) of cells before stopping transmitting. The same sequence of events occurs for stations {3} to {N ¹ 1}. In particular, when station {N ¹ 1} is satisfied, i.e. after the transmission of its quota of slots in state T, it sends the GNT upstream, transits to state FA and enters the non-restricted mode. At this point all the stations are in the same state as they were at the beginning of our observation. From the above considerations it follows that if the tagged station is always idle it observes cycles of busy slots of fixed duration V ¼ (Q r þ 2d)(N ¹ 1) þ Q s þ 1, as shown in Fig. 3. Hereafter, these busy slot cycles will be referred to as type A slot cycles. Without losing any generality we can assume the beginning of a cycle of busy slots at the instant at which station {N ¹ 1} observes the d-th slot after releasing the GNT (see Fig. 2). Thus, we can conclude that the slot pattern observed by the tagged station when it is always idle is cyclic, and cycle starting points occur at the instants at which the station {N ¹ 1} observes the d-th slot after the release of GNT. Let us assume that the tagged station is observing type A cycles, as shown in Fig. 3. As soon as the tagged station has cells to transmit, it immediately sends a REQ upstream since it is already starved and switches to state T.
Fig. 3. Slot occupancy patterns observed by the tagged station when idle.
Fig. 4. Type B slot cycle.
With reference to the instant at which station {N ¹ 1} observes the d-th after GNT release (which we assume as the initial instant of our analysis), the REQ can be sent by the tagged station in one of the two following non-overlapping time intervals: [¹2d,0) or [0,V ¹ 2d). The slot pattern observed by the tagged station varies depending on the interval. The two cases will be discussed separately below. 3.1.1. The REQ is sent by the tagged station in the interval [0,V ¹ 2d) In this case the REQ reaches station {N ¹ 1} after the station itself has released the GNT and has transmitted 2d slots in the non-restricted mode. However, the two following alternatives may occur. Either the station {N ¹ 1} has already sent a REQ upstream and, thus, is in state T in the restricted mode or, it is still in state FA and is observing busy slots. In the first case, it changes its state from T to B. In the latter case, it switches from state FA to H, continues to count busy slots until Q s þ 1, and then propagates the REQ upstream passing to state B. In any case the evolution of stations from {N ¹ 2} to {1} is not influenced by the fact that the tagged station has become active. Thus, station {j} (j ¼ 1, 2, …N ¹ 2) behaves exactly as shown in Fig. 2. On the other hand, the behaviour of station {N ¹ 1} is slightly different with respect to the case when the tagged station was idle. In fact, when station {N ¹ 1} becomes satisfied it can no longer forward the GNT upstream since it is now in state B. Therefore, it simply remains in state B and refrains from transmitting. Slots left empty by station {N ¹ 1} can be utilized by the tagged station which can transmit up to Q r cells. Specifically, the tagged station transmits cells until it is satisfied (transmission of Q r cells) or its queue is empty. When one of these two events occurs, the tagged station forwards the GNT upstream and changes its state from T to FA. Then it observes some more 2d empty slots before being covered by station {N ¹ 1}. Upon receiving the GNT, station {N ¹ 1} passes from state B to T and immediately after, since it is satisfied, to state FA, thus forwarding the GNT upstream. Stations {N ¹ 2} to {1} behave as they did when the tagged station was idle. The above considerations imply that when the tagged station sends a REQ in the interval [0,V ¹ 2d), immediately after V busy slots it observes Q þ 2d (Q # Qr ) empty slots (Fig. 4). Hereafter, the busy slot pattern shown in Fig. 4 will be referred to as type B slot cycle. In this slot cycle the former Q empty slots are observed by the tagged station when it is in the restricted mode, while the 2d empty slots are observed when it is in the non-restricted mode. At the time the tagged station observes the 2d ¹ th in the
G. Anastasi et al./Computer Communications 20 (1997) 671–680
Fig. 5. Type C slot cycle.
non-restricted mode the station {N ¹ 1} has already forwarded the GNT and is transmitting the d-th in the non-restricted mode. This means that from now on the tagged station again observes type A cycles until it sends a new REQ upstream. 3.1.2. The REQ is sent by the tagged station in the interval [¹2d,0) Suppose now that the REQ is sent by the tagged station inside a type A cycle in the interval [¹2d,0). It can be verified from Fig. 2 that the REQ reaches station {N ¹ 1} after station {N ¹ 1} itself has forwarded the GNT upstream and before it has completed the 2d-th transmission in the non-restricted mode. Upon receiving the REQ signal, station {N ¹ 1} changes its state from FA to B entering the restricted mode of operation. If the REQ is sent at instant ¹d, then 2d ¹ d is the number of cells transmitted by station {N ¹ 1} in the non-restricted mode at the reception of the REQ. Thus, station {N ¹ 1} will transmit more d cells in the restricted mode before being covered by traffic sent by station {N ¹ 2}. After observing Q s þ 1 busy slots it will forward the REQ upstream. The behaviour of stations {N ¹ 2} to {1} is not affected by the fact that the tagged station is now active. Thus, they still behave as shown in Fig. 2. Specifically, when station {N ¹ 2} becomes satisfied it refrains from transmitting and therefore after d time slots, station {N ¹ 1} again observes empty slots. Since it has already transmitted d cells in the restricted mode it will send only Q r ¹ d cells before stopping because it has been satisfied. Empty slots thus become available for the tagged station which, as in the previous case, transmits Q cells (Q # Qr ) in the non-restricted mode before forwarding the GNT upstream. Then it observes some more 2d empty slots before being covered by the traffic sent by station {N ¹ 1}. We can thus conclude that when the REQ is sent by the tagged station in the time interval [ ¹ 2d,0), it observes V ¹ d busy slots followed by Q þ 2d empty slots as shown in Fig. 5. As in the previous case, when the tagged station observes the last empty slot, station {N ¹ 1} is transmitting the d-th cell after it has forwarded the GNT upstream. This means that from now on the tagged station observes once more type A cycles, until it sends a new REQ upstream. Hereafter, a busy slot pattern of the type shown in Fig. 5 will be referred to as a type C slot cycle. As can be seen, type B and type C slot cycles only differ in terms of the number of busy slots. It is easy to verify that a type B or C cycle cannot be followed by a type C cycle. In fact, from the above
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considerations it follows that a type C cycle only occurs if the REQ is sent in the interval [¹2d,0) with respect to the beginning of the new cycle. On the other hand, if the tagged station observes a type B or C cycle, it sees empty slots in the interval [¹2d,0) and thus it cannot send a REQ. For the opposite reason a type B or C cycle may be followed by a type B cycle. In fact, during the interval [0,V ¹ 2d) (of a type B or C cycle) the tagged station observes busy slots and thus it may send a REQ. Remembering that a type A cycle can be followed by cycles of any type, the relationship between consecutive slot cycles can be summarized as follows. A → A, B, C B → A, B C → A, B The above considerations imply that the slot occupancy pattern observed by the tagged station is cyclic, and busy slot cycle starting points occur at instants at which station {N ¹ 1} completes the transmission of the d-th cell in the non-restricted mode. 3.2. Single server queue with vacation In order to simplify our analysis, we will maximize the number of busy slots in a type C slot cycle, V ¹ d, by V. This makes the analysis approximate but more conservative. However, since 0 # d # 2d, if the value of 2d is small compared with V the results provided by the approximate model are accurate1. As a consequence of the above approximation only types A and B slot cycles are observed by the tagged station. Since we are focusing on the tagged station, the system can be modelled by a single server queue with vacation. The service time is constant and equal to the slot duration while the server goes on vacation for a duration which is constant and equal to V slots. The number of consecutive service times is equal to Q þ 2d where Q # Qr .
4. Markov chain description We focus on the tagged station and observe the system immediately after the boundary of each slot (except for the last one) left empty by the station {N ¹ 1} (see Fig. 6). In addition, we also observe immediately after the (V ¹ 2d)-th busy slots (to establish whether or not the tagged station has sent a REQ). Hereafter, these particular instants will be referred to as embedding points. The state of the system at time n (n ¼ 0, 1, 2, …) is defined by the vector Y n ¼ [L,FA,S] n where L indicates the number of cells in the buffer of the tagged station at an embedding point, 1
The accuracy of the approximate model was checked by simulation.
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Fig. 6. Embedding points.
FA is a Boolean variable which indicates whether the tagged station is operating in the non-restricted mode (FA ¼ 1) or in the restricted mode (FA ¼ 0), S is the index of the embedding point within a busy slot cycle. The value of S depends on the value of FA. Specifically, if FA ¼ 0 (i.e. the tagged station operates in the restricted mode) then S [ [0,Q r], otherwise (FA ¼ 1)S [ [1,2d ¹ 1]. The system is thus described by the discrete time discrete state process Y ¼ {Y n, n [ N}. It can be verified that process Y is a Markov chain with state space E ¼ N 3 F where 8
2
ÿ
2
F ¼ (fa, s) : (fa ¼ 0) ∧ s [ 0, Qr ∨
2
ÿ
2
313
(fa ¼ 1)em ∧ ∫ [ ∞, ∈d ¹ ∞
3139
In order to establish the equations that describe the dynamics of L we will first analyse Y at the embedding point which is located immediately after the (V ¹ 2d)-th busy slot (S ¼ 0), and then at the other embedding points (S . 0) which immediately follow an empty slot. If S n þ1 ¼ 0 then the following relation holds: 8 Ln þ arr(V) > > <
Ln þ 1 ¼
> > :
arr(V ¹ 2d þ 1)
(
Ln þ 1 ¼
arr(1)
if Ln ¼ 0
(2)
Ln þ 1 ¼ Ln ¹ 1 þ arr(2d þ 1)
(3)
From the reported considerations and from an analysis of Fig. 3, the following relations related to the dynamics of FA and S can easily be derived.
FAn þ 1 ¼
8 > >0 > > > > 1 > > > > > <0
if Sn ¼ 0 if 0 , Sn , Qr , FAn ¼ 0 and Ln ¼ 0 if 0 , Sn , Qr , FAn ¼ 0 and Ln . 0
> 1 > > > > > > > 1 > > > :
if Sn ¼ Qr , FAn ¼ 0
Sn þ 1 ¼
if Sn ¼ 2d ¹ 1, FAn ¼ 1
8 > >0 > > > > 1 > > > > > < 1 > > > > > > > > > > > :
(4)
if 0 , Sn , 2d ¹ 1 and FAn ¼ 1
0
if Sn . 0 and Ln ¼ 0
where arr(T) indicates the number of arrivals in a time interval of duration equal to T slots. In fact, if S n ¼ 0 then the last slot cycle was of type A, and therefore the time interval between S n and S n þ1 is equal to the duration of a type A cycle (V). Furthermore, no cell has been transmitted between S n and S n þ1. If S n . 0 the last slot cycle was a type B cycle, and S n itself was located just after the penultimate slot of that cycle. Since S n þ1 is located immediately after the boundary of the (V ¹ 2d)-th busy slot of the new cycle, then the time interval between S n and S n þ1 is equal to V ¹ 2d þ 1 slots. Furthermore, if L n . 0 a cell has been transmitted in the last empty slot of the previous slot cycle. On the other hand, if S n þ1 . 0 we need to distinguish case S n þ1 . 1 from case S n þ1 ¼ 1. In the first case the temporal distance between S n and S n þ1 is always equal to one slot, and S n þ1 may or may not represent a cell departure instant depending on the state of buffer at time n. Hence
if Ln . 0
In the second case Eq. (2) still holds if FA n þ1 ¼ 1 (i.e. the tagged station operates in non-restricted mode), otherwise (FA n þ1 ¼ 0) the time interval between S n and S n þ1 is equal to 2d þ 1 slots (see Fig. 6) and the buffer contains at least one cell for which the REQ was sent. Thus it is
if Sn ¼ 0
Ln ¹ 1 þ arr(V ¹ 2d þ 1) if Sn . 0 and Ln . 0 (1)
Ln ¹ 1 þ arr(1)
if Sn ¼ 0 and Ln ¼ 0 if Sn ¼ 0 and L . 0 if Sn ¼ Qr and FAn ¼ 0
1
if Sn . 0, FAn ¼ 0 and Ln ¼ 0
0
if Sn ¼ 2d ¹ 1 and FAn ¼ 1
Sn þ 1
(5)
otherwise
Eqs (1)–(5) prove that process Y is an irreducible and aperiodic Markov chain. Furthermore, since at each embedding point the value of L decreases at the most by one unit, it follows that Y is a Markov chain of M/G/1-type [7] with level L and phase (FA,S). The transition probabilities matrix P is 0 1 Ln ¼ 0 B0 B1 B2 B3 B4 … B
Ln ¼ 1 B B A0
A1
A2
A3
A4
P ¼ Ln ¼ 2 B B 0
A0
A1
A2
A3
0
A0
A1
A2
…
…
…
…
B
B Ln ¼ 3 B @ 0
…
…
C C C …C C C …C A
…C
…
(6)
G. Anastasi et al./Computer Communications 20 (1997) 671–680
where A k and B k (k ¼ 0, 1, 2, …) are matrices of Q r þ 2d size. From Eqs (1)–(5) it is easy to verify that matrices A k and B k (k ¼ 0, 1, 2, …) have the following structure
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where a m indicates the average number of cells which arrive in m slots and a ¼ a 1. Under the hypothesis that a m ¼ ma, after some algebraic manipulations the stability condition
can be rewritten as follows r¼
V þ Qr þ 2d a,1 Qr þ 2d
(10)
Hence, (V þ Qr þ 2d)a , Qr þ 2d
where a(T) k indicates the probability that k cells arrive in a time interval of duration equal to T slots (T $ 0). Note that A k isXa circulant matrix and this implies that matrix ` A ¼ k ¼ 0 Ak iscirculant as well. If we calculate the vector X` T b¼ k ¼ 0 kAk · e where e ¼ [1, 1, …1] , then the stability condition requires that [7] r ¼ pb , 1
(7)
where vector p is the invariant probability vector of matrix A and thus satisfies the following matrix system (
p ¼ pA pe ¼ 1
(8)
Since A is a circulant matrix, vector p has all the elements equal to 1/(Q r þ 2d) where Q r þ 2d is the size of matrix A. From the structure of matrices A k it follows that in vector b the first component is the average number of cell arrivals in 2d þ 1 slots, the last component gives the average number of cells which arrive in V ¹ 2d þ 1 slots, and all the remaining Q r þ 2d ¹ 2 elements contain the average number of cell arrivals in one slot. Hence 1 [a þ (Qr þ 2d ¹ 2)a þ aV ¹ 2d þ 1 ] , 1 r¼ Qr þ 2d 2d þ 1
(11a)
Eq. (11a) can easily be interpreted. In fact, its left hand side gives the average number of cells which arrive in a type B cycle of maximum length (i.e. Q ¼ Q r). This number must be less than the number of empty slots observed in such a cycle. The stability condition can also be expressed in the following form a,
Qr þ 2d Qr þ 2d ¼ V þ Qr þ 2d (Qs þ 1) þ N(Qr þ 2d)
(11b)
It can be proved that the quantity (Qr þ 2d)= ((Qs þ 1) þ N(Qr þ 2d)) represents the normalized throughput that the tagged station would achieve if it operated in asymptotic conditions [1]. This means that as r approaches unity, the influence of the approximation introduced in Section 3.2 becomes negligible. In fact, when r increases, all slot cycles tend to be of type B.
5. Solution of the model Following the standard methodology for the solution of an M/G/1-type Markov chain, our first step was to calculate the G matrix which satisfies the following matrix equation G¼
` X
Ak Gk
(12)
k¼0
(9)
Then we derived vector x 0 which is proportional X` to the stationary probability vector of matrix K ¼ k ¼ 0 Bk Gk
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and is normalized by x 0k ¼ 1 [7,4] where k¼eþ
` X
Bi
i¼1 ` X
g¼ I ¹
i¼1
iX ¹1
Gk g
(13)
k¼0
Ai
iX ¹1
!¹1 k
G
J[F
e
(14)
k¼0
Once x 0 has been calculated, by applying Ramaswami’s algorithm [7,10] we obtain the probability vectors "
xk ¼ x0 B¯ k þ
kX ¹1
the buffer at any embedding point or after the last empty slots of a cycle, and that this instant is a cell departure instant, is given by X pk ¼ pk [J] þ zk k ¼ 0, 1, 2, … (20)
#
xj A¯ k þ 1 ¹ j (I ¹ A¯ 1 ) ¹ 1 , k $ 1
Hence, by normalizing p k (k ¼ 0, 1, 2,…) we obtain the conditional probability of having k cells in the buffer of the tagged station at cell departure instants p9k ¼
(15)
pk ` X
k ¼ 0, 1, 2, …
(21)
pk
k¼0
j¼1
where A¯ n ¼
` X
Ai Gi ¹ n and B¯ n ¼
i¼n
` X
6. Results
Bi Gi ¹ n for n $ 0:
i¼n
The generic element of x k, x k[J], k $ 0, J [ F, gives the probability that, in steady state conditions, the Markov chain is at level L ¼ k and with phase (fa,s) ¼ J, i.e. the probability of having k cells in the buffer of the tagged station at the embedding point identified by couple (fa,s) ¼ J. It can be written as xk [fa, s] ¼ pk [fa, s] þ p¯ k [fa, s]
(16)
In our analysis we assume that the arrival process is Poisson. In accordance with Burke’s and PASTA theorems under this assumption Eq. (21) gives the probability mass function of the buffer occupancy at any instant. Hence, we can derive the average queue length at the tagged station and, by applying Little’s theorem, the average access delay experienced by a cell at the tagged station. Observe that by defining m¼
where pk [fa, s] (p¯ k [fa, s]) is the probability that k cells are in the buffer at the embedding point identified by couple (fa,s) and that the embedding point is (is not) a cell departure instant. Hence, pk [fa, s] ¼ xk [fa, s] ¹ p¯ k [fa, s]
(17)
It can be verified that
p¯ k [fa, s] ¼
8 x0 [fa, s] > > > > > > 0 > > < > > > > > > > > :
Qr X
x0 [0, t]·a(1) k
V þ Qr þ 2d Qr þ 2d
then Eq. (10) implies that r ¼ am. Hence, r represents the coefficient of utilization of the system, and is a function of the offered load to the system. Fig. 7 reports the average access (queuing þ service) delay experienced by cells transmitted by the tagged station as a function of the coefficient of utilization r for a given set of network parameter values which are summarized in Table 1. The average access delay remains almost constant
if fa ¼ 0 and s ¼ 0 if fa ¼ 0 and s [ [1Qr ] if fa ¼ 1 and s ¼ 1
t¼1
x0 [1, s ¹ 1]·a(1) k
if fa ¼ 1 and s [ [2, 2d ¹ 1] (18)
By substituting Eq. (18) into Eq. (17) the probability vectors p k, k ¼ 1, 1, 2…, can easily be calculated. Let us observe that the probability that k (k ¼ 0, 1, 2,…) cells are in the buffer immediately after the last empty slot observed by the tagged station and that this slot has been utilized by the tagged station itself is zk ¼
kX þ1
xj [1, 2d ¹ 1]·a(1) k ¹ j þ 1 k ¼ 0, 1, 2, …
(19)
j¼1
Thus, the probability p k (k ¼ 0, 1, 2,…) of having k cells in
Fig. 7. Average access delay vs. coefficient of utilization.
G. Anastasi et al./Computer Communications 20 (1997) 671–680
Fig. 8. Influence of the number of interfering stations on the average access delay.
(less than or equal to 100 slots < 300 ms) up to r ¼ 0.9. Subsequently, as the r value increases it dramatically increases. This is due to the long duration of the vacation interval (V ¼ 176 slots). In fact, at least for r # 0.9 the delay experienced by a cell in the buffer of the tagged station is mainly due to the fact that the tagged station observes busy slots. Only when r approaches 1 is the dominant contribution in the average access delay given by the queuing delay. Obviously, the average access delay experienced by cells transmitted by the tagged station depends on the number N of interfering stations. In fact, as N increases, the vacation period increases as well. Fig. 8 shows the influence of parameter N on the average access delay. For each value of N the shape of the curve is the same as in the case where N ¼ 5. Fig. 9 shows the dependency of the average access delay on the Q r value. As above, an increase in the Q r value implies a longer vacation period and thus a greater average access delay. Hence, a low Q r value is recommended. However, this conclusion contrasts with the results obtained in [1] where it was proved that when all the stations operate in asymptotic conditions the network fairness (expressed in terms of the throughput achieved by each station) increases as the ratio Q r/Q s increases.
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Fig. 9. Influence of the Q r value on the average access delay.
7. Conclusions In this paper we have proposed and solved a worst-case model of the MetaRing MAC protocol implementing a local fairness algorithm. The analysis focused on the average access delay experienced by cells transmitted by a tagged station. Under the assumption of Poisson arrivals the following results were obtained. The average access delay remains almost constant for values of the coefficient of utilization r # 0.9. Then, there is a sharp increase as r approaches 1. The average access delay increases with the number of interfering stations N and with the protocol parameter Q r. Due to the dependency of the average access delay on Q r, a low value for Q r is advisable. However, in [1] it was proved that when all the stations operate in asymptotic conditions the network fairness increases with the ratio Q r/Q s. The analysis proposed in this paper can be enhanced by considering an arrival process which models a real arrival process better than the Poisson process. This will be the next step in our study.
References Table 1 Network parameter values Capacity of each ring Slot size Slot duration Distance between stations (d) Number of interfering stations (N) Starvation quota (Q s) Transmission quota (Q r)
150 Mbps 53 octets 2.82 ms 7 slots < 4 Km 5 3 30
[1] G. Anastasi, M. La Porta, L. Lenzini, A performance study of the local fairness algorithm for the MetaRing MAC protocol, submitted for publication. [2] J. Chen, I. Cidon, Y. Ofek, A local fairness algorithm for gigabit LAN’s/MAN’s with spatial reuse, IEEE J. Sel. Area Commun. 11 (8) (1993) 1183–1192. [3] I. Cidon, Y. Ofek, MetaRing, a full duplex ring with fairness and spatial reuse, IEEE Trans. Commun. 41(1) (1993). [4] G. Latouche, Newton’s iteration for non-linear equations in Markov chains, IMA J. Numer. Anal. 14 (1994) 583–598.
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[5] M. La Porta, Valutazione delle prestazioni del meccanismo di fairness locale REQ/GNT applicato alla MAN al gigabit/sec MetaRing, Laurea Thesis (in Italian). [6] A. Mayer, Y. Ofek, M. Yung, Approximating max-min rates via distributed local scheduling with partial information, private communication. [7] M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and their Applications, Marcel Dekker, New York, 1989. [8] Y. Ofek, Overview of the MetaRing architecture, Comp. Networks ISDN Syst. 26 (6–8) (1994) 817–830. [9] H.T. Wu, Y. Ofek, K. Sorhaby, Integration of synchronous and asynchronous traffic on the MetaRing architecture and its analysis, Proceedings of ICC ’92. [10] V. Ramaswami, A stable recursion for the steady state vector in Markov chains of M/G/1-type, Stoch. Mod. 4 (1988) 183–188.
Giuseppe Anastasi received the ‘Laurea’ degree in Electronic Engineering and the PhD degree in Computer Engineering from the University of Pisa (Italy) in 1990 and 1995, respectively. In 1991 he joined the Department of Information Engineering of the University of Pisa where he is currently a research fellow. His scientific interests include wireless and mobile networks, high speed networks, service integration, modelling and performance evaluation of computer networks. He is a member of the IEEE Computer Society.
Marco La Porta graduated in computer Engineering from the University of Pisa (Italy) in 1995. He is currently the system manager of the Computer Laboratory of the Faculty of Engineering of the University of Pisa. His research interests include LANs, MANs, design, modelling and performance evaluation of computer networks.
Luciano Lenzini holds a degree in Physics from the University of Pisa, Italy. He joined CNUCE, and institute of the Italian National Research Council (CNR) in 1970. Starting in 1973, he spent a year and a half at the IBM Scientific Centre in Cambridge, Massachusetts, working on computer networks. He has since directed several national and international projects some of which are in order: RPCNET, the first Italian packet switching network; STELLA, the first European broadcasting satellite network and OSIRIDE, the first Italian OSI network. His current research interests include integrated service networks, the design and performance evaluation of both Metropolitan Area Network MAC protocols and packet-based radio access mechanisms for third generation mobile systems.
Computer Communications 20 (1997) 671–680
Delay analysis of a worst-case model of the MetaRing MAC protocol with local fairness G. Anastasi*, M. La Porta, L. Lenzini University of Pisa, Department of Information Engineering, Via Diotisalvi 2, 56126 Pisa, Italy Received 3 November 1995; accepted 16 February 1996
Abstract The MetaRing is a medium access control (MAC) protocol for gigabit LANs and MANs with cells removed by the destination stations (slot reuse). Slot reuse increases the aggregate throughput beyond the capacity of single links but may cause starvation. In order to prevent this the MetaRing MAC protocol includes a fairness mechanism. Two types of fairness algorithms have been proposed: ‘global’ and ‘local’. The MetaRing analysed in this paper implements the local fairness algorithm specified in [2]. In order to reduce the complexity we have identified a simplified model which can be analytically solved and yet still provides useful information on network performances. This model represents a worst-case scenario in which network congestion is stressed, i.e. no station, apart from a specific station (tagged station), ever has an empty queue. The model proposed can be represented by a discrete time discrete state Markov chain of M/G/1-type and hence the matrix analytical methodology has been used to solve it. Our analysis focuses on the average access delay experienced by the tagged station as a function of the offered load. The results show that average access delay remains bounded for values of offered load less than or equal to 90%. Furthermore, the average access delay depends on the number of interfering stations and on the protocol parameter Q r. q 1997 Elsevier Science B.V. Keywords: MetaRing; Gigabit MANs; Local fairness
1. Introduction The MetaRing is a MAC protocol for high speed LANs and MANs with spatial reuse, which can achieve an aggregate throughput much higher than the network capacity [8]. However, spatial reuse may cause starvation at stations covered by upstream highly loaded stations. A fairness mechanism is thus required to prevent this from occurring. Fairness protocols have already been proposed and they can be classified into ‘global’ [3] and ‘local’ [2,6]. A globalfairness algorithm regulates access to the shared media by considering the network as a single communication resource. By contrast, a local- fairness algorithm views the media as a distributed collection of communication resources. Hence, a local fairness algorithm regulates the transmissions of the interfering stations without affecting the others so that local fairness algorithms can exploit the advantages of spatial reuse better than global algorithms. A simulative analysis of a MetaRing MAC protocol with a local fairness algorithm was performed in [2], where the * Corresponding author. Tel.: +39 50 568511; fax: +39 50 568522; e-mail [email protected]
0140-3664/97/$17.00 q 1997 Elsevier Science B.V. All rights reserved PII S 01 40 - 36 6 4( 9 7) 0 00 5 6- X
network was analysed in asymptotic conditions (i.e. when stations always have traffic to send). Station and aggregate throughputs were measured for various network scenarios. The results reported in [2] show that the throughput achieved by a station is not influenced by the behaviour of stations which do not interfere with it. Thus, stations in the networks can be divided into disjoint groups. Within a single group the throughput achieved by each station, as well as the aggregate throughput (i.e. the sum of the throughputs of all the stations in the group), strictly depends on the interactions among stations belonging to the group. In [1] we focused on a network scenario which is realistic but particularly critical with respect to slot reuse. It consists of a single group of N þ 1 stations located in one half of the ring. The interference between stations is maximized since stations {1} to {N} are assumed to send traffic only to station {N þ 1} which operates as a gateway to an external network (Fig. 1). The analysis was divided into two parts. In the first part, the network was analysed in asymptotic conditions, and closed formulae for the aggregate and station throughputs were derived. In the second part, a thorough simulation analysis of the network was performed in underload (i.e.
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G. Anastasi et al./Computer Communications 20 (1997) 671–680
Fig. 1. Network scenario.
when the offered load is lower than the network capacity) and overload (i.e. when the offered load is greater than the network capacity) conditions. The rationale behind the simulative choice was that the modelling and performance analysis of the MetaRing MAC protocol with local fairness is very difficult. In fact, an exact model of the network should take into consideration interactions among the stations in the network. This makes an analysis almost impossible. To overcome these difficulties in the present paper we identify a simplified model of the network scenario shown in Fig. 1. This model can be analytically solved and yet provides useful information on network performance. This model represents a worst-case scenario in which network congestion is stressed (i.e. no station, apart from a specific station, hereafter referred to as the tagged station, ever has empty queue). We will refer to this model as the worst-case model, which will be analysed using the embedded Markov chain technique. We will show that the transition probability matrix of this embedded Markov chain is of M/G/1-type [7]. We will thus apply the matrix analytic methodology to calculate the stationary probabilities of the embedded Markov chain. This paper is organized as follows. Section 2 outlines the MetaRing MAC protocol and the local fairness algorithm. Section 3 describes the worst-case model. Section 4 is devoted to the solution of the model, while the results obtained are reported and discussed in Section 5. Conclusions are drawn in Section 6.
2. The MetaRing MAC protocol with local fairness The MetaRing is a medium access control (MAC) protocol for LANs and MANs operating at a speed of 1 Gbit/s and above [8]. It connects a set of stations by means of a bi-directional ring made up of full duplex point-to-point serial links. The protocol provides two types of services, asynchronous and synchronous [9]. This paper only deals with the asynchronous type. A MetaRing network can operate under two basic access control modes: buffer insertion for variable size packets, and slotted for
fixed length cells. In this paper only the slotted access mode is considered. According to the slotted access mode, information is segmented into cells and each cell transmitted in one slot. Slots are structured into a header and an information field. The header includes a busy bit which indicates whether the slot is empty (busy bit ¼ 0) or busy. A cell can be accommodated on an empty slot. Since the ring is bi-directional the MetaRing MAC protocol uses the shortest path criterion to choose one of the possible directions. Cells are removed by the destination station which frees the slot by resetting the busy bit to zero. After cells have been removed, the slot can be reused by the same station or by its downstream stations. Slot reuse increases the aggregate throughput beyond the capacity of single links but may cause starvation. To prevent this from occurring a fairness mechanism is required. The fairness algorithm analysed in this paper is local and is triggered by any station whenever it foresees a potential starvation. The algorithm is based on two control signals: REQ (REQuest) and GNT (GraNT) which propagate in the opposite direction with respect to the information flow they regulate. According to this algorithm a station can be in a finite set of states and, correspondingly, in two different operation modes, non-restricted and restricted. In the former there are no limitations on the number of cells a station can transmit, while in the latter a station can only send a predefined quota, Q r, of cells. Note that in [2] the condition under which a station establishes that it will become starved is not specified. Therefore, to carry out our analysis we had to integrate the local fairness algorithm specified in [2] with the following definition: a station observing more than Q s contiguous busy slots becomes starved. Clearly, other definitions are possible and each one may lead to different performances. With our definition of starved station, the local fairness algorithm works as follows. If no conflict occurs (i.e. if no transmitting station is covered by the traffic originated by upstream stations) all the stations are in the free access (FA) state. In this state a station operates in non-restricted mode and therefore it can transmit a cell every time it observes an empty slot. Once a station (say station {i}) becomes starved, it triggers the local fairness mechanism by sending the REQ signal to the upstream station {i ¹ 1} and then enters the tail (T) state. Upon reception of a REQ, station {i ¹ 1} enters the restricted mode of operation and the head (H) state. Then, if station {i ¹ 1} itself observes busy slots from upstream stations it immediately forwards the REQ to station {i ¹ 2} and enters the body (B) state. While in states T, B and H a station can send at most a quota Q r of cells (restricted mode). Upon satisfaction (i.e. transmission of the predefined quota) or as soon as the buffer becomes empty, the tail station sends a GNT signal to its upstream station and switches back to the non-restricted FA state. Upon receiving this GNT, the upstream station follows similar rules: if it is
G. Anastasi et al./Computer Communications 20 (1997) 671–680
in state B, it transits to state T and will forward the GNT upon satisfaction or when its buffer becomes empty (whichever occurs first). If a station is in state H, it switches to state FA and the local fairness cycle involving a set of stations located on that ring segment is terminated. The local fairness mechanism thus creates a request path which contains unique and distinct head and tail stations for each segment in the ring where there are interfering stations. Furthermore, since there might be multiple initiators of the fairness algorithm (and, therefore, of a request path), two or more distinct request paths may overlap. When this occurs the overlapping request paths are merged into a unique request path. Further details on the local fairness algorithm can be found in [2].
3. Worst-case model description To reduce the complexity of the MetaRing MAC protocol modelling, we focus on station {N} (tagged station) and we assume that the remaining stations (i.e. stations {1}, {2}, …, {N ¹ 1}) operate in asymptotic conditions (i.e. they always have cells to transmit). Furthermore, we assume that the distance between stations {j} and {j þ 1} (j ¼ 1, 2, …, N ¹ 1) is constant and equal to an integer number d of slots. Thus, we consider a worst-case model which is analytically tractable and yet provides useful information on network performance. We begin the description of the worst-case model by
673
analyzing the slot occupancy pattern observed by the tagged station. In order to make this easier we first derive the slot occupancy pattern by assuming that the tagged station is idle (it never has traffic to send). Then, we investigate how the slot occupancy pattern varies when the tagged station becomes active. 3.1. Slot occupancy pattern observed by the tagged station Under the hypothesis that the tagged station is always idle the network operates in asymptotic conditions and, thus, the considerations reported in [1] can be applied. Specifically, the network behaviour is cyclic and can be represented by means of Fig. 2, which describes the operations performed by each station. Without losing any generality, we can observe the evolution of the network since the instant at which the station {N ¹ 1} forwards the GNT signal to the upstream station entering state FA. It can be shown that at this instant all the upstream stations are satisfied [5]. Hence, when station {j} (j ¼ N ¹ 2, N ¹ 3, …, 2) receives the GNT from a downstream station, it switches from state B to T and, immediately after, sends the GNT. Then, it transits to state FA and enters the non-restricted mode. The behaviour of station {1} when GNT arrives is slightly different from the other stations. Since this station is the head of the request path it immediately transits to state FA. The above considerations imply that, after GNT has been released, station {j} (j ¼ N ¹ 1, N ¹ 2, …, 2) observes a
Fig. 2. Station operations when the tagged station is idle.
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G. Anastasi et al./Computer Communications 20 (1997) 671–680
train of 2d empty slots. Since {j} operates in non-restricted mode it uses all of them before being covered by the upstream stations. Now, let us analyse the REQ propagation. When station {N ¹ 1} is covered by the upstream traffic it counts for Q s þ 1 busy slots and then performs the following operations: it sends a REQ upstream, switches to state T and enters the restricted mode of operation. Stations numbered from {N ¹ 2} to {2} all execute the same operations. Each of them receives the REQ from the downstream station and at the same time each observes the (Q s þ 1)-th consecutive busy slot (and hence they would be ready to send a REQ upstream). The simultaneous occurrence of the two events is managed by each of these stations as follows: they first move from state FA to H and, immediately after, forward the REQ upstream entering state B. The trajectory of the REQ is thus parallel to the trajectory of the GNT as shown in Fig. 2. Since station {N ¹ 1} issues a REQ 2d þ Q s þ 1 time slots after GNT release, the REQ itself reaches station {1} exactly 2d þ Q s þ 1 slots after GNT arrival at station {1} itself. This implies that station {1} transmits 2d þ Q s þ 1 cells in state FA before receiving a REQ from the downstream station. Upon reception of the REQ, station {1} changes its state from FA to H, enters the restricted mode, and sends Q r more cells before stopping transmitting because it has been satisfied. After a time interval equal to d slots, station {2} observes empty slots and thus it sends its quota (Q r) of cells before stopping transmitting. The same sequence of events occurs for stations {3} to {N ¹ 1}. In particular, when station {N ¹ 1} is satisfied, i.e. after the transmission of its quota of slots in state T, it sends the GNT upstream, transits to state FA and enters the non-restricted mode. At this point all the stations are in the same state as they were at the beginning of our observation. From the above considerations it follows that if the tagged station is always idle it observes cycles of busy slots of fixed duration V ¼ (Q r þ 2d)(N ¹ 1) þ Q s þ 1, as shown in Fig. 3. Hereafter, these busy slot cycles will be referred to as type A slot cycles. Without losing any generality we can assume the beginning of a cycle of busy slots at the instant at which station {N ¹ 1} observes the d-th slot after releasing the GNT (see Fig. 2). Thus, we can conclude that the slot pattern observed by the tagged station when it is always idle is cyclic, and cycle starting points occur at the instants at which the station {N ¹ 1} observes the d-th slot after the release of GNT. Let us assume that the tagged station is observing type A cycles, as shown in Fig. 3. As soon as the tagged station has cells to transmit, it immediately sends a REQ upstream since it is already starved and switches to state T.
Fig. 3. Slot occupancy patterns observed by the tagged station when idle.
Fig. 4. Type B slot cycle.
With reference to the instant at which station {N ¹ 1} observes the d-th after GNT release (which we assume as the initial instant of our analysis), the REQ can be sent by the tagged station in one of the two following non-overlapping time intervals: [¹2d,0) or [0,V ¹ 2d). The slot pattern observed by the tagged station varies depending on the interval. The two cases will be discussed separately below. 3.1.1. The REQ is sent by the tagged station in the interval [0,V ¹ 2d) In this case the REQ reaches station {N ¹ 1} after the station itself has released the GNT and has transmitted 2d slots in the non-restricted mode. However, the two following alternatives may occur. Either the station {N ¹ 1} has already sent a REQ upstream and, thus, is in state T in the restricted mode or, it is still in state FA and is observing busy slots. In the first case, it changes its state from T to B. In the latter case, it switches from state FA to H, continues to count busy slots until Q s þ 1, and then propagates the REQ upstream passing to state B. In any case the evolution of stations from {N ¹ 2} to {1} is not influenced by the fact that the tagged station has become active. Thus, station {j} (j ¼ 1, 2, …N ¹ 2) behaves exactly as shown in Fig. 2. On the other hand, the behaviour of station {N ¹ 1} is slightly different with respect to the case when the tagged station was idle. In fact, when station {N ¹ 1} becomes satisfied it can no longer forward the GNT upstream since it is now in state B. Therefore, it simply remains in state B and refrains from transmitting. Slots left empty by station {N ¹ 1} can be utilized by the tagged station which can transmit up to Q r cells. Specifically, the tagged station transmits cells until it is satisfied (transmission of Q r cells) or its queue is empty. When one of these two events occurs, the tagged station forwards the GNT upstream and changes its state from T to FA. Then it observes some more 2d empty slots before being covered by station {N ¹ 1}. Upon receiving the GNT, station {N ¹ 1} passes from state B to T and immediately after, since it is satisfied, to state FA, thus forwarding the GNT upstream. Stations {N ¹ 2} to {1} behave as they did when the tagged station was idle. The above considerations imply that when the tagged station sends a REQ in the interval [0,V ¹ 2d), immediately after V busy slots it observes Q þ 2d (Q # Qr ) empty slots (Fig. 4). Hereafter, the busy slot pattern shown in Fig. 4 will be referred to as type B slot cycle. In this slot cycle the former Q empty slots are observed by the tagged station when it is in the restricted mode, while the 2d empty slots are observed when it is in the non-restricted mode. At the time the tagged station observes the 2d ¹ th in the
G. Anastasi et al./Computer Communications 20 (1997) 671–680
Fig. 5. Type C slot cycle.
non-restricted mode the station {N ¹ 1} has already forwarded the GNT and is transmitting the d-th in the non-restricted mode. This means that from now on the tagged station again observes type A cycles until it sends a new REQ upstream. 3.1.2. The REQ is sent by the tagged station in the interval [¹2d,0) Suppose now that the REQ is sent by the tagged station inside a type A cycle in the interval [¹2d,0). It can be verified from Fig. 2 that the REQ reaches station {N ¹ 1} after station {N ¹ 1} itself has forwarded the GNT upstream and before it has completed the 2d-th transmission in the non-restricted mode. Upon receiving the REQ signal, station {N ¹ 1} changes its state from FA to B entering the restricted mode of operation. If the REQ is sent at instant ¹d, then 2d ¹ d is the number of cells transmitted by station {N ¹ 1} in the non-restricted mode at the reception of the REQ. Thus, station {N ¹ 1} will transmit more d cells in the restricted mode before being covered by traffic sent by station {N ¹ 2}. After observing Q s þ 1 busy slots it will forward the REQ upstream. The behaviour of stations {N ¹ 2} to {1} is not affected by the fact that the tagged station is now active. Thus, they still behave as shown in Fig. 2. Specifically, when station {N ¹ 2} becomes satisfied it refrains from transmitting and therefore after d time slots, station {N ¹ 1} again observes empty slots. Since it has already transmitted d cells in the restricted mode it will send only Q r ¹ d cells before stopping because it has been satisfied. Empty slots thus become available for the tagged station which, as in the previous case, transmits Q cells (Q # Qr ) in the non-restricted mode before forwarding the GNT upstream. Then it observes some more 2d empty slots before being covered by the traffic sent by station {N ¹ 1}. We can thus conclude that when the REQ is sent by the tagged station in the time interval [ ¹ 2d,0), it observes V ¹ d busy slots followed by Q þ 2d empty slots as shown in Fig. 5. As in the previous case, when the tagged station observes the last empty slot, station {N ¹ 1} is transmitting the d-th cell after it has forwarded the GNT upstream. This means that from now on the tagged station observes once more type A cycles, until it sends a new REQ upstream. Hereafter, a busy slot pattern of the type shown in Fig. 5 will be referred to as a type C slot cycle. As can be seen, type B and type C slot cycles only differ in terms of the number of busy slots. It is easy to verify that a type B or C cycle cannot be followed by a type C cycle. In fact, from the above
675
considerations it follows that a type C cycle only occurs if the REQ is sent in the interval [¹2d,0) with respect to the beginning of the new cycle. On the other hand, if the tagged station observes a type B or C cycle, it sees empty slots in the interval [¹2d,0) and thus it cannot send a REQ. For the opposite reason a type B or C cycle may be followed by a type B cycle. In fact, during the interval [0,V ¹ 2d) (of a type B or C cycle) the tagged station observes busy slots and thus it may send a REQ. Remembering that a type A cycle can be followed by cycles of any type, the relationship between consecutive slot cycles can be summarized as follows. A → A, B, C B → A, B C → A, B The above considerations imply that the slot occupancy pattern observed by the tagged station is cyclic, and busy slot cycle starting points occur at instants at which station {N ¹ 1} completes the transmission of the d-th cell in the non-restricted mode. 3.2. Single server queue with vacation In order to simplify our analysis, we will maximize the number of busy slots in a type C slot cycle, V ¹ d, by V. This makes the analysis approximate but more conservative. However, since 0 # d # 2d, if the value of 2d is small compared with V the results provided by the approximate model are accurate1. As a consequence of the above approximation only types A and B slot cycles are observed by the tagged station. Since we are focusing on the tagged station, the system can be modelled by a single server queue with vacation. The service time is constant and equal to the slot duration while the server goes on vacation for a duration which is constant and equal to V slots. The number of consecutive service times is equal to Q þ 2d where Q # Qr .
4. Markov chain description We focus on the tagged station and observe the system immediately after the boundary of each slot (except for the last one) left empty by the station {N ¹ 1} (see Fig. 6). In addition, we also observe immediately after the (V ¹ 2d)-th busy slots (to establish whether or not the tagged station has sent a REQ). Hereafter, these particular instants will be referred to as embedding points. The state of the system at time n (n ¼ 0, 1, 2, …) is defined by the vector Y n ¼ [L,FA,S] n where L indicates the number of cells in the buffer of the tagged station at an embedding point, 1
The accuracy of the approximate model was checked by simulation.
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Fig. 6. Embedding points.
FA is a Boolean variable which indicates whether the tagged station is operating in the non-restricted mode (FA ¼ 1) or in the restricted mode (FA ¼ 0), S is the index of the embedding point within a busy slot cycle. The value of S depends on the value of FA. Specifically, if FA ¼ 0 (i.e. the tagged station operates in the restricted mode) then S [ [0,Q r], otherwise (FA ¼ 1)S [ [1,2d ¹ 1]. The system is thus described by the discrete time discrete state process Y ¼ {Y n, n [ N}. It can be verified that process Y is a Markov chain with state space E ¼ N 3 F where 8
2
ÿ
2
F ¼ (fa, s) : (fa ¼ 0) ∧ s [ 0, Qr ∨
2
ÿ
2
313
(fa ¼ 1)em ∧ ∫ [ ∞, ∈d ¹ ∞
3139
In order to establish the equations that describe the dynamics of L we will first analyse Y at the embedding point which is located immediately after the (V ¹ 2d)-th busy slot (S ¼ 0), and then at the other embedding points (S . 0) which immediately follow an empty slot. If S n þ1 ¼ 0 then the following relation holds: 8 Ln þ arr(V) > > <
Ln þ 1 ¼
> > :
arr(V ¹ 2d þ 1)
(
Ln þ 1 ¼
arr(1)
if Ln ¼ 0
(2)
Ln þ 1 ¼ Ln ¹ 1 þ arr(2d þ 1)
(3)
From the reported considerations and from an analysis of Fig. 3, the following relations related to the dynamics of FA and S can easily be derived.
FAn þ 1 ¼
8 > >0 > > > > 1 > > > > > <0
if Sn ¼ 0 if 0 , Sn , Qr , FAn ¼ 0 and Ln ¼ 0 if 0 , Sn , Qr , FAn ¼ 0 and Ln . 0
> 1 > > > > > > > 1 > > > :
if Sn ¼ Qr , FAn ¼ 0
Sn þ 1 ¼
if Sn ¼ 2d ¹ 1, FAn ¼ 1
8 > >0 > > > > 1 > > > > > < 1 > > > > > > > > > > > :
(4)
if 0 , Sn , 2d ¹ 1 and FAn ¼ 1
0
if Sn . 0 and Ln ¼ 0
where arr(T) indicates the number of arrivals in a time interval of duration equal to T slots. In fact, if S n ¼ 0 then the last slot cycle was of type A, and therefore the time interval between S n and S n þ1 is equal to the duration of a type A cycle (V). Furthermore, no cell has been transmitted between S n and S n þ1. If S n . 0 the last slot cycle was a type B cycle, and S n itself was located just after the penultimate slot of that cycle. Since S n þ1 is located immediately after the boundary of the (V ¹ 2d)-th busy slot of the new cycle, then the time interval between S n and S n þ1 is equal to V ¹ 2d þ 1 slots. Furthermore, if L n . 0 a cell has been transmitted in the last empty slot of the previous slot cycle. On the other hand, if S n þ1 . 0 we need to distinguish case S n þ1 . 1 from case S n þ1 ¼ 1. In the first case the temporal distance between S n and S n þ1 is always equal to one slot, and S n þ1 may or may not represent a cell departure instant depending on the state of buffer at time n. Hence
if Ln . 0
In the second case Eq. (2) still holds if FA n þ1 ¼ 1 (i.e. the tagged station operates in non-restricted mode), otherwise (FA n þ1 ¼ 0) the time interval between S n and S n þ1 is equal to 2d þ 1 slots (see Fig. 6) and the buffer contains at least one cell for which the REQ was sent. Thus it is
if Sn ¼ 0
Ln ¹ 1 þ arr(V ¹ 2d þ 1) if Sn . 0 and Ln . 0 (1)
Ln ¹ 1 þ arr(1)
if Sn ¼ 0 and Ln ¼ 0 if Sn ¼ 0 and L . 0 if Sn ¼ Qr and FAn ¼ 0
1
if Sn . 0, FAn ¼ 0 and Ln ¼ 0
0
if Sn ¼ 2d ¹ 1 and FAn ¼ 1
Sn þ 1
(5)
otherwise
Eqs (1)–(5) prove that process Y is an irreducible and aperiodic Markov chain. Furthermore, since at each embedding point the value of L decreases at the most by one unit, it follows that Y is a Markov chain of M/G/1-type [7] with level L and phase (FA,S). The transition probabilities matrix P is 0 1 Ln ¼ 0 B0 B1 B2 B3 B4 … B
Ln ¼ 1 B B A0
A1
A2
A3
A4
P ¼ Ln ¼ 2 B B 0
A0
A1
A2
A3
0
A0
A1
A2
…
…
…
…
B
B Ln ¼ 3 B @ 0
…
…
C C C …C C C …C A
…C
…
(6)
G. Anastasi et al./Computer Communications 20 (1997) 671–680
where A k and B k (k ¼ 0, 1, 2, …) are matrices of Q r þ 2d size. From Eqs (1)–(5) it is easy to verify that matrices A k and B k (k ¼ 0, 1, 2, …) have the following structure
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where a m indicates the average number of cells which arrive in m slots and a ¼ a 1. Under the hypothesis that a m ¼ ma, after some algebraic manipulations the stability condition
can be rewritten as follows r¼
V þ Qr þ 2d a,1 Qr þ 2d
(10)
Hence, (V þ Qr þ 2d)a , Qr þ 2d
where a(T) k indicates the probability that k cells arrive in a time interval of duration equal to T slots (T $ 0). Note that A k isXa circulant matrix and this implies that matrix ` A ¼ k ¼ 0 Ak iscirculant as well. If we calculate the vector X` T b¼ k ¼ 0 kAk · e where e ¼ [1, 1, …1] , then the stability condition requires that [7] r ¼ pb , 1
(7)
where vector p is the invariant probability vector of matrix A and thus satisfies the following matrix system (
p ¼ pA pe ¼ 1
(8)
Since A is a circulant matrix, vector p has all the elements equal to 1/(Q r þ 2d) where Q r þ 2d is the size of matrix A. From the structure of matrices A k it follows that in vector b the first component is the average number of cell arrivals in 2d þ 1 slots, the last component gives the average number of cells which arrive in V ¹ 2d þ 1 slots, and all the remaining Q r þ 2d ¹ 2 elements contain the average number of cell arrivals in one slot. Hence 1 [a þ (Qr þ 2d ¹ 2)a þ aV ¹ 2d þ 1 ] , 1 r¼ Qr þ 2d 2d þ 1
(11a)
Eq. (11a) can easily be interpreted. In fact, its left hand side gives the average number of cells which arrive in a type B cycle of maximum length (i.e. Q ¼ Q r). This number must be less than the number of empty slots observed in such a cycle. The stability condition can also be expressed in the following form a,
Qr þ 2d Qr þ 2d ¼ V þ Qr þ 2d (Qs þ 1) þ N(Qr þ 2d)
(11b)
It can be proved that the quantity (Qr þ 2d)= ((Qs þ 1) þ N(Qr þ 2d)) represents the normalized throughput that the tagged station would achieve if it operated in asymptotic conditions [1]. This means that as r approaches unity, the influence of the approximation introduced in Section 3.2 becomes negligible. In fact, when r increases, all slot cycles tend to be of type B.
5. Solution of the model Following the standard methodology for the solution of an M/G/1-type Markov chain, our first step was to calculate the G matrix which satisfies the following matrix equation G¼
` X
Ak Gk
(12)
k¼0
(9)
Then we derived vector x 0 which is proportional X` to the stationary probability vector of matrix K ¼ k ¼ 0 Bk Gk
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and is normalized by x 0k ¼ 1 [7,4] where k¼eþ
` X
Bi
i¼1 ` X
g¼ I ¹
i¼1
iX ¹1
Gk g
(13)
k¼0
Ai
iX ¹1
!¹1 k
G
J[F
e
(14)
k¼0
Once x 0 has been calculated, by applying Ramaswami’s algorithm [7,10] we obtain the probability vectors "
xk ¼ x0 B¯ k þ
kX ¹1
the buffer at any embedding point or after the last empty slots of a cycle, and that this instant is a cell departure instant, is given by X pk ¼ pk [J] þ zk k ¼ 0, 1, 2, … (20)
#
xj A¯ k þ 1 ¹ j (I ¹ A¯ 1 ) ¹ 1 , k $ 1
Hence, by normalizing p k (k ¼ 0, 1, 2,…) we obtain the conditional probability of having k cells in the buffer of the tagged station at cell departure instants p9k ¼
(15)
pk ` X
k ¼ 0, 1, 2, …
(21)
pk
k¼0
j¼1
where A¯ n ¼
` X
Ai Gi ¹ n and B¯ n ¼
i¼n
` X
6. Results
Bi Gi ¹ n for n $ 0:
i¼n
The generic element of x k, x k[J], k $ 0, J [ F, gives the probability that, in steady state conditions, the Markov chain is at level L ¼ k and with phase (fa,s) ¼ J, i.e. the probability of having k cells in the buffer of the tagged station at the embedding point identified by couple (fa,s) ¼ J. It can be written as xk [fa, s] ¼ pk [fa, s] þ p¯ k [fa, s]
(16)
In our analysis we assume that the arrival process is Poisson. In accordance with Burke’s and PASTA theorems under this assumption Eq. (21) gives the probability mass function of the buffer occupancy at any instant. Hence, we can derive the average queue length at the tagged station and, by applying Little’s theorem, the average access delay experienced by a cell at the tagged station. Observe that by defining m¼
where pk [fa, s] (p¯ k [fa, s]) is the probability that k cells are in the buffer at the embedding point identified by couple (fa,s) and that the embedding point is (is not) a cell departure instant. Hence, pk [fa, s] ¼ xk [fa, s] ¹ p¯ k [fa, s]
(17)
It can be verified that
p¯ k [fa, s] ¼
8 x0 [fa, s] > > > > > > 0 > > < > > > > > > > > :
Qr X
x0 [0, t]·a(1) k
V þ Qr þ 2d Qr þ 2d
then Eq. (10) implies that r ¼ am. Hence, r represents the coefficient of utilization of the system, and is a function of the offered load to the system. Fig. 7 reports the average access (queuing þ service) delay experienced by cells transmitted by the tagged station as a function of the coefficient of utilization r for a given set of network parameter values which are summarized in Table 1. The average access delay remains almost constant
if fa ¼ 0 and s ¼ 0 if fa ¼ 0 and s [ [1Qr ] if fa ¼ 1 and s ¼ 1
t¼1
x0 [1, s ¹ 1]·a(1) k
if fa ¼ 1 and s [ [2, 2d ¹ 1] (18)
By substituting Eq. (18) into Eq. (17) the probability vectors p k, k ¼ 1, 1, 2…, can easily be calculated. Let us observe that the probability that k (k ¼ 0, 1, 2,…) cells are in the buffer immediately after the last empty slot observed by the tagged station and that this slot has been utilized by the tagged station itself is zk ¼
kX þ1
xj [1, 2d ¹ 1]·a(1) k ¹ j þ 1 k ¼ 0, 1, 2, …
(19)
j¼1
Thus, the probability p k (k ¼ 0, 1, 2,…) of having k cells in
Fig. 7. Average access delay vs. coefficient of utilization.
G. Anastasi et al./Computer Communications 20 (1997) 671–680
Fig. 8. Influence of the number of interfering stations on the average access delay.
(less than or equal to 100 slots < 300 ms) up to r ¼ 0.9. Subsequently, as the r value increases it dramatically increases. This is due to the long duration of the vacation interval (V ¼ 176 slots). In fact, at least for r # 0.9 the delay experienced by a cell in the buffer of the tagged station is mainly due to the fact that the tagged station observes busy slots. Only when r approaches 1 is the dominant contribution in the average access delay given by the queuing delay. Obviously, the average access delay experienced by cells transmitted by the tagged station depends on the number N of interfering stations. In fact, as N increases, the vacation period increases as well. Fig. 8 shows the influence of parameter N on the average access delay. For each value of N the shape of the curve is the same as in the case where N ¼ 5. Fig. 9 shows the dependency of the average access delay on the Q r value. As above, an increase in the Q r value implies a longer vacation period and thus a greater average access delay. Hence, a low Q r value is recommended. However, this conclusion contrasts with the results obtained in [1] where it was proved that when all the stations operate in asymptotic conditions the network fairness (expressed in terms of the throughput achieved by each station) increases as the ratio Q r/Q s increases.
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Fig. 9. Influence of the Q r value on the average access delay.
7. Conclusions In this paper we have proposed and solved a worst-case model of the MetaRing MAC protocol implementing a local fairness algorithm. The analysis focused on the average access delay experienced by cells transmitted by a tagged station. Under the assumption of Poisson arrivals the following results were obtained. The average access delay remains almost constant for values of the coefficient of utilization r # 0.9. Then, there is a sharp increase as r approaches 1. The average access delay increases with the number of interfering stations N and with the protocol parameter Q r. Due to the dependency of the average access delay on Q r, a low value for Q r is advisable. However, in [1] it was proved that when all the stations operate in asymptotic conditions the network fairness increases with the ratio Q r/Q s. The analysis proposed in this paper can be enhanced by considering an arrival process which models a real arrival process better than the Poisson process. This will be the next step in our study.
References Table 1 Network parameter values Capacity of each ring Slot size Slot duration Distance between stations (d) Number of interfering stations (N) Starvation quota (Q s) Transmission quota (Q r)
150 Mbps 53 octets 2.82 ms 7 slots < 4 Km 5 3 30
[1] G. Anastasi, M. La Porta, L. Lenzini, A performance study of the local fairness algorithm for the MetaRing MAC protocol, submitted for publication. [2] J. Chen, I. Cidon, Y. Ofek, A local fairness algorithm for gigabit LAN’s/MAN’s with spatial reuse, IEEE J. Sel. Area Commun. 11 (8) (1993) 1183–1192. [3] I. Cidon, Y. Ofek, MetaRing, a full duplex ring with fairness and spatial reuse, IEEE Trans. Commun. 41(1) (1993). [4] G. Latouche, Newton’s iteration for non-linear equations in Markov chains, IMA J. Numer. Anal. 14 (1994) 583–598.
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[5] M. La Porta, Valutazione delle prestazioni del meccanismo di fairness locale REQ/GNT applicato alla MAN al gigabit/sec MetaRing, Laurea Thesis (in Italian). [6] A. Mayer, Y. Ofek, M. Yung, Approximating max-min rates via distributed local scheduling with partial information, private communication. [7] M.F. Neuts, Structured Stochastic Matrices of M/G/1 Type and their Applications, Marcel Dekker, New York, 1989. [8] Y. Ofek, Overview of the MetaRing architecture, Comp. Networks ISDN Syst. 26 (6–8) (1994) 817–830. [9] H.T. Wu, Y. Ofek, K. Sorhaby, Integration of synchronous and asynchronous traffic on the MetaRing architecture and its analysis, Proceedings of ICC ’92. [10] V. Ramaswami, A stable recursion for the steady state vector in Markov chains of M/G/1-type, Stoch. Mod. 4 (1988) 183–188.
Giuseppe Anastasi received the ‘Laurea’ degree in Electronic Engineering and the PhD degree in Computer Engineering from the University of Pisa (Italy) in 1990 and 1995, respectively. In 1991 he joined the Department of Information Engineering of the University of Pisa where he is currently a research fellow. His scientific interests include wireless and mobile networks, high speed networks, service integration, modelling and performance evaluation of computer networks. He is a member of the IEEE Computer Society.
Marco La Porta graduated in computer Engineering from the University of Pisa (Italy) in 1995. He is currently the system manager of the Computer Laboratory of the Faculty of Engineering of the University of Pisa. His research interests include LANs, MANs, design, modelling and performance evaluation of computer networks.
Luciano Lenzini holds a degree in Physics from the University of Pisa, Italy. He joined CNUCE, and institute of the Italian National Research Council (CNR) in 1970. Starting in 1973, he spent a year and a half at the IBM Scientific Centre in Cambridge, Massachusetts, working on computer networks. He has since directed several national and international projects some of which are in order: RPCNET, the first Italian packet switching network; STELLA, the first European broadcasting satellite network and OSIRIDE, the first Italian OSI network. His current research interests include integrated service networks, the design and performance evaluation of both Metropolitan Area Network MAC protocols and packet-based radio access mechanisms for third generation mobile systems.