Deadlock-Free Information Structure Distributed Mutual Exclusion Algorithms

Proceedings of the 19th ACSC Conference,Melbourne, Australia, January 31 - February 21996.Deadlock-Free Information Structure DistributedMutual Exclusion AlgorithmsUmberto Bonollo Elizabeth A. SonenbergDepartment of Computer Science The University of MelbourneParkville 3052Australia umberto@cs.mu.oz.au eas@cs.mu.oz.au AbstractIn the distributed mutual exclusion problem, a set of processes must coordinate their actions so that at any time at most one process may be within its critical section. Information structure distributed mutual exclusion algorithms (ISDME) are instances of a general algorithm which can represent several non-token-based algorithms. This paper introduces a new deadlock-free ISDME algorithm (DF-ISDME)which operates on a restricted class of information structures. DF-ISDME allows deadlock-free solutions for a wider class of information structure topologies than a previously identified class of deadlock-free Maekawa algorithms (DF-Maekawa),but retains the same throughput as DF-Maekawa.The paper provides a detailed description of the DF-ISDME algorithm, and correctness proofs.Message performance of the new algorithm is discussed, as are potential application scenarios.Keywords Distributed systems, distributed algorithms, distributed mutual exclusion, operating systems.1IntroductionIn the distributed mutual exclusion problem [2, 6,10], a set of distributed processes must coordinate their actions such that at any time at most one process has access to a shared resource: for example,a network printer, or a tuple in a distributed database.A process may only be within its critical section (CS)whenever it has gained exclusive access to the shared resource. Processes can only communicate with each other by message passing - there is no shared memory.There are several well known distributed mutual exclusion algorithms which have been published in the literature. Generally, these distributed mutual exclusion algorithms can be classed as either token-based or information structure (ISDME)algorithms. In token-based algorithms, a unique token is passed among the processes. Possession of the token guarantees mutual exclusion. The token-based algorithm of Suzuki and Kasami [11] is a well known example. Information structure algorithms are instances of a general algorithm (GEN-ISDME)proposed by Sanders [7] as a means of classifying several non-token-based algorithms [2, 6, 3]. Among the advantages of information structure algorithms is their potential for local handling of process failure and process recovery while, by comparison, token-based algorithms require global cooperation for handling process failure which can be quite complex [7]. A taxonomy of distributed mutual exclusion algorithms appears in the survey by Singhal [9].A distributed system is a set of distributed processes, i n ∈{,,...,}12, which are connected by a communications network. Most descriptions of distributed mutual exclusion algorithms assume a specific communications and processing environment with the following properties: Processes may be asynchronous meaning that there are time intervals within which no instructions need be executed; Each process is associated with a communication channel from which it can receive messages and each process can send messages to the communication channels of any of the other processes; Communication between the processes is assumed to be reliable meaning that all messages sent are eventually delivered, and that messages can't be corrupted or duplicated; Message transit times are finite but arbitrary; Messages between any two processes are assumed to arrive in the order they were sent, called first-in-first-out(FIFO) order; Message arrivals at a process are handled serially; and finally processes can't fail. The performance of a mutual exclusion algorithm can be measured by the average number of messages (M )exchanged per execution of a CS and the synchronisation delay (D ) between successive executions of the CS in response to concurrent requests for CS access.In an ISDME algorithm, each process has a pair of sets called the request set and the inform set which form the local information structure . The collection of these pairs defines the global information structure. For the GEN-ISDME algorithm, each process requests permission to enter its CS by sending Request messages to the processes which are members of its request set 1 followed by waiting for Grant messages to be received from them. Upon exiting its CS, the process will then send Release messages to the processes which are members of its inform set. Mutual exclusion is only guaranteed when the request sets and the inform sets satisfy a certain condition on the global information structure.It turns out that the request sets and inform sets can describe the topology of several well known and practical mutual exclusion algorithms [7] including the following which are listed here along with their message performance M : The Centralised Algorithm [4] with a single centralised coordinator, M =3; The fully-distributed Ricart-Agrawala [6] Algorithm,M n =−21(); Lamport [2] Fully Distributed Algorithm, M n =−31(); and Maekawa [3]Algorithm, O (n ). While there are algorithms which can enter the CS by sending very few messages (e.g., O (log n ) [5]), this is also accompanied by longer delays D to enter the CS (e.g.also O (log n ) [5]). On the other hand, most ISDME algorithms have delays of D =1 [6] or of D =2 [3]which means they perform well under heavy demand for the CS.Unfortunately, due to arbitrary communication delays, the GEN-ISDME algorithm can deadlock under certain information structures [7] and so is impractical. Sanders proposed a deadlock-detecting algorithm (DDGEN-ISDME) which detects and avoids potential deadlock situations: it uses a deadlock recovery scheme requiring the exchange of Inquire , Yield , and Fail messages. The deadlock recovery scheme is quite complex. As Sanders [7] has pointed out, the original version of the Maekawa algorithm used such a scheme but didn't send Fail messages under certain circumstances hence still leaving a scenario for deadlock. The potential for deadlock means that ISDME algorithms may perform1 It's crucial that the Request messages are sent without waiting for replies and in no particular order;Request messages could be implemented using some broadcast primitive.worse than their lower bound whenever the potential for deadlock arises. The probability of the lower bound not being achieved increases under conditions of heavy demand for the CS.The objective of this paper is to identify a class of deadlock-free ISDME algorithms which do not require any deadlock recovery scheme. As we will show, this will be achieved by introducing a new deadlock-free ISDME algorithm (DF-ISDME) which operates on a more restricted class of information structures than the class introduced by Sanders for GEN-ISDME. Our DF-ISDME algorithm only has a synchronisation delay of D =1 which is twice as fast as the delay of the Maekawa algorithm D =2 [3].Research related to this work has been carried out by Singhal [8]. Singhal proposed an algorithm for a class of deadlock-free Maekawa algorithms (DF-Maekawa) whose class of information structures is a restriction on the class admitted by the standard Maekawa algorithms. The message performance for DF-Maekawa is 312()n − compared with O (n )for Maekawa as the deadlock freedom is achieved at the expense of increased communication costs, but the delay is D =1. The DF-ISDME algorithm introduced here can handle a more general class of information structures than DF-Maekawa can yet it still retains the D =1 delay. The class of information structures handled by DF-ISDME may be viewed along a continuum from DF-Maekawa on the one hand to Ricart-Agrawala [6] on the other. However,our focus is based on the information structure approach of Sanders [7] whereas Singhal's focus in [8] was solely on the classes admitted by Maekawas's algorithm [3].In the following sections, we describe the DF-ISDME algorithm. We then present formal proofs of its correctness. The performance of the algorithm under optimal conditions is shown We show an example of its application. We then conclude with our discussion.2 A Deadlock-Free ISDME AlgorithmIn this section, we define the concept of information structures for ISDME algorithms and describe a new deadlock-free ISDME algorithm (DF-ISDME).2.1Information StructuresThe following definitions apply to information structures in an ISDME algorithm: The local i nformation structure for a process i is defined by a pair of sets called, the request set , R i , and the inform set , I i . The global information structure for a set of processes is the collection of local information structures for each process, i n ∈{,,...,}12. A process i must send a Request message before entering the CS to all the processes which are members of its requestset, R i and it must only send Release messages on exiting the CS to the processes which are members of its inform set, I i. For convenience, we define for each process i the status set, St i. The status set St i is derived from the inform sets of the processes using the following rule,∀ij:i I j∈↔j St i∈. A process i keeps track of any processes j St i∈since j must inform i before j enters the CS and after j exits the CS.Figures 1 and 2 show two ISDME algorithms whose topology is described in terms of their global information structures. The local information structure of each process together with its corresponding status set is also shown. As in [7], for processes i and j, a solid arc from i to j indicates that j I i∈ and j R i∈. A dashed arc from i to j indicates that j R i∈but j I i∉. Figure 1 corresponds to the symmetric fully-distributed Ricart-Agrawala algorithm [6] in which each process requests permission from every other process before entering the CS. Figure 2 corresponds to an algorithm in which each process requests permission from a predetermined subset of processes. In Figure 2, either process 2 or process 3 will request entry by asking processes 1, 2, and 3; process 1 only asks itself, but receives requests from the other two processes; when process 2 or process 3 exits the CS it must inform process 1 by sending it a Release message; processes 2 and 3 never send Release messages to each other.Beverly Sanders introduced a general algorithm called DDGEN-ISDME in [7] which guarantees mutual exclusion if a global information structure satisfies the following information structure conditions.ISDME Mutual Exclusion Theorem (Sanders [7]). The information structure conditions ∀∈i i I i:,∀⊆i I Ri i:, and ∀≠ij i j:()I Ii j∩≠∅∨()i R j Rj i∈∧∈are necessary and sufficient conditions for the DDGEN-ISDME algorithm to guarantee mutual exclusion.For mutual exclusion, the DF-ISDME algorithm requires that the following information structure conditions hold: ∀∈i i I i:, ∀⊆i I Ri i:, and ∀≠ij i j: ()i I j Ij i∈∨∈∨()i R j Rj i∈∧∈(For a proof see Section 3.2). The class of information structures allowed by the DF-ISDME information structure conditions is a subset of the class of information structures allowed by Sanders' ISDME theorem [7]. The DDGEN-ISDME algorithm requires a complex deadlock-recovery scheme for deadlock-freedom [7, 8]. The DF-ISDME algorithm is deadlock-free but does not require deadlock-recovery. Both of the algorithms shown in Figures 1 and 2 are deadlock-free with respect to the DF-ISDME algorithm.2.2DF-ISDME Algorithm Description In this section we present the DF-ISDME algorithm code. The following variables are used by a process i: The request set R i and the inform set I i which form the local information structure together with the status set St i derived from I i. A logical clock clk [2] which is used to tag each Request message sent by i with a logical time t such that the next value of t will be one greater than the highest time value that i has received thus far. A priority request queue Q i which maintains an ascending order sorted list of the timestamps associated with Request messages received by process i. A timestamp is a tuple consisting of the time value of a Request together with the process identifier of the process making the request. A timestamp (,)i t i is less than (has higher priority)than a timestamp (,)j t j, written (,)(,)i t j ti j<, when ()()t t t t i ji j i j<∨=∧<holds. The function priority_insr(Q i,(,)j t j) returns a new priority queue inserting the timestamp (,)j t j into the priority request queue described by the current valueof Q i. Send statements are of the form send j msgi(,) which sends the message represented by the tuple msg to the channel associated with process j.Send statements are always enabled when control reaches them and they never block. Receive statements are ofR2 = {1,2,3,4} I1 = {1}I2 = {2}St1 = {1}St2 = {2}R3 = {1,2,3,4} I4 = {4}I3 = {3}St4 = {4}St3 = {3} Figure 1: Ricart-Agrawala ISDME InformationStructure.R1 = {1}R2 = {1,2,3} I1 = {1}I2 = {1,2}St2 = {2}I3 = {1,3}St3 = {3}Figure 2: Example ISDME Information Structure.the form recv msgi() which receives the next message from the channel associated with process i into the tuple described by msg.Receive statements are enabled for a process i when control reaches them and the channel associated with i is not empty, otherwise the receive statement will block until a message for i arrives in the channel.The DF-ISDME algorithm for a process i is defined by the following program code:loop// process i waiting (requesting) to enter the CSw1:clk := clk + 1w2:t := clkw3:∀s ∈ R i : send i( s, (Req, i, t, clk) )w4:grants_left := | R i |w5:while grants_left > 0 ∨ (i, t) ≠ head(Q i) doh1:recv i( (msg, r, u, c) )h2:clk := max(clk, c)h3:if msg = Req then // Request msg, u=crq1:Q i := priority_insr(Q i, (r, u))rq2:if¬( (,)(,)i t r u< ) thenrq3:send i( r, (Gr, i, u, clk) )rq4:if r ∉ St i thenrq5:Q i := Q i - {(r, u)}endifendifelseh4:if msg = Rel then // Rel msg: only if i ∈ I j rl1:Q i := Q i - {(r, u)}e lseh5:if msg = Gr then // Grant messagegr1:grants_left := grants_left - 1endif // Grendif // Relendif // Reqendwhile// process i can now enter the CScs:doCS// process i exiting the CSex1:∀s ∈ I i : send i( s, (Rel, i, t, clk) )ex2:Remain i := Q iex3:while Remain i≠φdorm1:(r, v) := head(Remain i)rm2:send i( r, (Gr, i, v, clk) )rm3:if r ∉ St i thenrm4:Q i := Q i - {(r, v)}endifrm5:Remain i := tail(Remain i)endwhileendloopFrom the program code, a process i enters the waiting region, w1..w5,chooses a time value t, and tries to gain permission to enter the CS by sending Request (Req) messages to all the members of its request set, R i. Process i's current request to enter is noted in t to be compared against any other requesting processes. Whenever a process i receives a Request message (see h1..h3) from a process j, it inserts j's request timestamp into Q i. Now if i's pending request t is of lower priority (higher time value) than j's then i will immediately send a Grant (Gr) message to j.At this point ( see rq4..rq5), j's timestamp can be deleted from Q i but only if it is not a member of i's status set St i. If i's pending request t is not of lower priority than j's then i will retain j's timestamp in Q i and the Grant message to j will be delayed. Process i will only reach the CS after it has received Grant messages from all the processes in its request set R i(grants_left=0) and its own request timestamp is at the head of Q i. On leaving the CS, a process i enters the exit region, ex1..ex3,and sends Release (Rel) messages to all the processes in its inform set, I i (For a process j I i∈, this will have the effect of removing i's timestamp, see rl1,from Q j). Now, process i has had its turn and can immediately send Grant messages to the processes in Q i(whose requests were delayed). Again, at this point ( see rm3..rm4), j's timestamp can be deleted from Q i but only if it is not a member of i's status set St i.One of the intuitions behind the algorithm is that process i must retain in Q i any timestamp (,)j t j whenever j St i∈since i is being informed that j is wishing to enter the CS, and it need not be the case, from the DF-ISDME information structure conditions above, that i will necessarily make a request on j.3Proof of CorrectnessIn this section, we present proofs for mutual exclusion, deadlock freedom and no individual starvation for DF-ISDME. The form of the latter two proofs is similar to Singhal [8].3.1Preliminary DefinitionsIn this section, we present several definitions to be used in the accompanying proofs of the DF-ISDME algorithm.As discussed previously, For a process i, the status set St i is derived from the inform sets of the processes using the following rule,∀ij:i I j∈↔j St i∈ (1).Propositions which are state predicates of the form at_l where l is a label like cs or ex1hold at a state whenever control is at a location which is infront of the program statement labelled by l.Propositions of the form in_li..j are used to denote that control is at or within a range of locations labelled l l i i ,+1, ..., l j .A timestamp (,)i t i is less than (has higher priority) than a timestamp (,)j t j , written (,)(,)i t j t i j <, when ()()t t t t i j i j i j <∨=∧< holds.The proposition, granted ij means that "process i has been granted permission to enter the CS with respect to process j ." More formally, the meaning of granted ij depends on whether j is a member of i 's inform set, define granted ij as j ∈ I i : (¬sent_rl ij since sent_gr ji ) ∨ j ∈ R i - I i : (¬in_ex1..3i since sent_gr ji ) (2) which says that if j is a member of i 's inform set then i has been granted permission by j whenever i has not yet sent a Release message to j since the last time that j sent i a Grant message, or if j is not a member of i 's inform set (but still in i's request set) then i has been granted permission by j whenever i has not yet entered its exit section (labelled by ex1 to ex3 in the DF-ISDME algorithm)since the last time that j sent i a Grant message.The proposition granted i means that "process i has been granted [full] permission to enter the CS."More formally, granted i ≡ ∀k ∈ R i : granted ik ∧(i, t i ) = head (Q i ) (3), which says that a process i has been granted permission to enter the CS if and only if all processes k which are members of i 's request set have granted their permission and the timestamp (i, t i ) is at the head of i 's priority queue Q i .3.2Mutual ExclusionDF-ISDME Mutual Exclusion Theorem.The information structure conditions ∀∈i i I i : (4),∀⊆i I R i i : (5), and ∀≠ij i j : ()i I j I j i ∈∨∈ ∨i R j R j i ∈∧∈ (6) are sufficient conditions for the DF-ISDME algorithm to guarantee mutual exclusion.Proof.The safety proof will be by induction on the execution sequences of the distributed processes.With respect to the DF-ISDME algorithm, we claim the following formulas are invariant:at cs i _→granted i (7), at cs j _→ granted j (8),granted i j ij ,≠→(,)(,)i t j t i j <(9),and¬∧(__)at cs at cs i j (10). Invariant (7) states that if control for i reaches the location in the algorithm labelled by cs i then i must have been granted permission to enter the CS. Similarly, (8) for j.Invariant (9) states that the only situation in which i can be granted permission by j is when i 's request timestamp has higher priority than j 's request timestamp. Invariant (10) is the mutual exclusionrequirement stating that no two processes i and j can both be in the CS at the same time.Invariance of (7) and (8).For i, the only transition that could affect the truth of (7) is w5⇒cs . Here at cs i _ will be true but from the while-loop transition grants left i _=0 ∧(,)()i t head Q i i = must hold which implies granted i so the invariant still holds. Thus (7) is invariant and so too is (8) for j. oInvariance of (9).For i and j, we will focus on when the antecedent granted ij ,i ≠j becomes true and argue that the consequent is also true. When i 's request is received by j , i 's timestamp will be compared at rq2. The only way a Grant message will be sent to i after rq3 is if the if-statement condition holds which means (,)(,)i t j t i j < holds where u = t j . Thus (9) is invariant.oInvariance of (10) (Mutual Exclusion Requirement).Without loss of generality, the only way (10) can be violated is if i is in the CS (at_cs i holds) and j enters via the transition w cs j j 5⇒.1. Hence, assume that both i and j have entered the CS (at_cs i ∧ at_cs j ) and show that a contradiction will arise.2. From 1. and the invariance of (7) and (8), it follows that granted i ∧ granted j is true; recall from (3) that granted i ≡ ∀k ∈ R i : granted ik ∧(i, t i ) = head (Q i ), and similarly for j.3. Assume that the information structure conditions, (4), (5), and (6), hold, then it follows that i R j ∈∨j R i ∈ which means that at least one of the processes must send a Request message to the other process.4. Say, (without loss of generality) that i asks j then by (7) j must have granted that request,granted ij .5. From 4. the premise of (9) must hold and hence (,)(,)i t j t i j <.6. From 3. and 4. there are two cases: (i) j ∈ I i or (ii) j R I i i ∈−.7. Case 6. (i): here j ∈ I i and then from (1)i St j ∈, and so (,)i t i must be a member of the priority queue, Q j .8. From 7. and from 5. it follows that (,)i t i will be ahead of (,)j t j in the priority queue, Q j which is a contradiction with 2. granted j which states that (,)j t j will be at the head of Q j .9. Case 6. (ii): here j R I i i ∈− (and then from (1)i St j ∉), and to maintain the assumption in 3. itfollows that i I j∈∨()i R j Rj i∈∧∈. So j must ask (send a Request message to) i here in addition to the assumption from 4. that i asks j.10. From 4. and 5. it is still the case that (,)(,)i t j ti j<since granted ij, and in 9. we have argued that in addition j must ask i, but in this case i will not grant j's request since the if-statement transition condition labelled by rq i2is false, so ¬<((,)(,))j t i tj i→¬granted ji, contrapositive of (9). This is in contradiction to 2. granted j and hence granted ji. n3.3Deadlock FreedomDeadlock could occur for a set of processes if they were each involved in a circular wait. A circular wait could occur if each of the processes i in the cycle is blocked at the location labelled by h i1 and is yet to receive a Grant message from the successor process in the cycle and no there are no Grant messages in transit which are destined for any of these processes. Assume, by way of contradiction, that this is the case. Then each process in the circular wait has delayed sending a Grant message to its predecessor process in the cycle. A process i will only defer sending a Grant to a process j if it is in the region w1..w5and (,)(,)i t j ti j<. Thus, to achieve a deadlock, each process in the circular wait must be blocked by a process with a higher priority request which is impossible. Therefore, the algorithm is deadlock-free.3.4No Individual StarvationStarvation occurs when a few processes repeatedly execute their CSs while other processes wait indefinitely in the region w1..w5. Assume, by way of contradiction, that process j has been repeatedly executing its CS while process i has been waiting in the region w1..w5.Given that the information structure conditions, (4), (5), and (6) of the DF-ISDME mutual exclusion theorem hold, then it follows that i R j∈∨j R i∈ which means that at least one of the processes must send a Request message to the other process. Thus, we have the following two cases:Case 1. i R j∈.Process j asks i for permission to enter the CS which if j has priority will result in i sending a Grant message to j.At h2,after j has processed i's Grant message, its next request to execute the CS will get a timestamp lower in priority than the pending request of i. So the next request of j cannot have priority over the pending request of i and therefore i will not send a Grant message to j unless it has executed its CS.Case 2. j R i∈.Process j can repeatedly execute the CS as long as i's request hasn't reached it yet. However, at h2, after j has received i's Request message, j's next request to execute the CS will get a timestamp lower in priority than the pending request of i. Therefore, j will send a Grant message to i before its next execution of the CS.Therefore, contrary to our assumption a process cannot repeatedly execute its CS while other processes are waiting for their turn to execute the CS. Hence there is no individual starvation.4PerformanceIn this section, we consider the optimal conditions for message performance and the synchronisation delay for the DF-ISDME algorithm.4.1Message PerformanceThe mean message performance M is the average number of messages required by a single process per execution of the CS. From Figure 2, if there is a solid arc from a process i to a process j then three messages will be required along the arc per CS execution, that is a Request message will be sent to j, a Grant message will be received from j,and a Release message will be sent from i to j, since j I i∈which satisfies the inform-set part of the information structure condition (6). From Figure 2, if there is a dashed arc from a process i to a process j then two messages will be required along the arc per CS execution, that is a Request message will be sent to j, and a Grant message will be received from j,since j R i∈which satisfies the request-set part of the information structure condition (6). Let a represent the number of dashed arcs between any pair of distinct processes i and j. Let b represent the number of solid arcs between any pair of distinct processes i and j;self-loop solid arcs are not counted2. Therefore, the mean message performance M for a DF-ISDME algorithm is given by M a b n=+()23 (11).To find the optimal mean message performance, we need to build the minimum request and inform sets that will satisfy the DF-ISDME mutual exclusion theorem. We note that to satisfy the theorem, it is only necessary to have a single solid arc either from i to j or from j to i. Thus, in abstract, we can view any optimal information structure as an undirected graph consisting of either one single solid arc (in some direction) or consisting of a dashed arc (which2Since any messages that a process sends to itself need not incur a transmission cost in a practical implementation.represents the pair of directed dashed arcs from i to j and from j to i ). Hence if a and b have their meaning as before with b restricted to only count one occurrence of a solid arc between processes i and j,then the following relationship holds:n n a b ()()−=+122 (12). We note that n n ()−12is the number of edges in a graph where each node has one edge to every other node in the graph. Now using (12) and substituting for a in the expression for message performance (11) above, we obtain M n b n =−−21() with 0≤≤b n n ()−12 (13).Now, M is smallest when b =n n ()−1, and therefore M n =−321() which is equivalent to the DF-Maekawa algorithm [8]. Conversely, M is largest when b =0, and therefore M n =−21() which is equivalent to the Ricart-Agrawala algorithm [6].Thus the optimal message performance of DF-ISDME lies along a continuum from DF-Maekawa on the one hand to Ricart-Agrawala on the other depending on b the number of edges which are single solid arcs in the information structure.4.2Synchronisation DelayThe synchronisation delay D is the maximum number of sequential message propagations from one process to another between successive executions of the CS. It is assumed that the next process to enter the critical section is blocked and only waiting for the process in the CS to exit. For the DF-ISDME algorithm the delay is D =1 since the next process j to enter is either waiting to receive a Release message to remove the timestamp ahead of its own request in Q j or (,)j t j is at the head of Q j and it is waiting to receive a Grant message to proceed. The DF-ISDME algorithm has twice the throughput of the Maekawa [3] algorithm which has a delay of D =2. The DF-ISDME algorithm retains the D =1 delay of the DF-Maekawa algorithm [8].5ApplicationIn this section, we show the potential application of the DF-ISDME algorithm. As an application example, consider that telecommunication network topologies may sometimes span administrative management domains such as for international calls between countries. Further suppose that half of the processes are colocated in one domain and half in the other domain. Since these management domains are managed by separate organisations, it may be the case that strictly optimal topologies may not be cost-effective or achievable. For example, although an M n =−321/() topology is the lower bound for DF-ISDME and equivalent to DF-Maekawa [8], it achieves this average performance for a pair of processes by asking one of them to send threemessages and the other none, yielding an average of 32 messages per process - it also assumes that costs can be amortised across management domains. Now,if one of the processes must originate a connection to a process in the other management domain then the originating process might need to hold the call open for the duration of the execution of its CS because it needs to send a Release message on exiting the CS. It may be much more cost-effective to allow processes that communicate between management domains to send 2 messages each per process prior to CS entry with no further communication required on CS exit.6DiscussionThe class of information structures which satisfies the DF-ISDME mutual exclusion theorem is a subset of the class of information structures which satisfies the ISDME mutual exclusion theorem of Sanders [7].However, the class of DF-ISDME information structures is a superset of the class of information structures admitted by DF-Maekawa [8]. The DF-ISDME algorithm has a synchronisation delay of D =1 which is twice as fast as the Maekawa algorithm [3] which has a delay of D =2. The DF-ISDME algorithm still retains the D =1 delay of the DF-Maekawa algorithm [8]. The DF-ISDME algorithm may be viewed as representing a continuum from DF-Maekawa on the one hand to Ricart-Agrawala [6] on the other. As Singhal [8] has shown for the DF-Maekawa algorithm (and hence for the lower bound of DF-ISDME), the message performance of M n =−321() is better than the worst case 5n of the Maekawa algorithm for n ≤17 under situations of heavy demand for the CS.The DF-ISDME algorithm allows deadlock-free solutions to apply to a wider class of information structure topologies than the DF-Maekawa algorithm.The move from deadlock-prone Maekawa [3] to deadlock-free DF-ISDME means that the message performance has gone from O (n ) to O (n ). The deadlock-freedom has been achieved at the expense of increased communication costs. The information structure approach used here is interesting because it provides an abstract view of an entire class of algorithms. We observe that a deadlock-free solution not requiring deadlock-recovery is impossible for certain information structures which satisfy the ISDME mutual exclusion theorem of Sanders [7](like Maekawa [3]) but do not satisfy the DF-ISDME theorem presented here.One of the motivations behind the work in this paper is to better understand the impact of global system requirements such as deadlock-freedom on the knowledge (and hence communication) required by the processes in a distributed system. Our current research work is using a knowledge-theoretic。