February 2012
Abstract: Udding's solution as presented in his paper and in our class is correct (i.e., starvation-free). However, Udding's wording of what constitutes a weak semaphore needs to be improved.
1 P(eu); ne := ne+1; V(eu); 2 P(qu); 3 P(eu); 4 nm := nm+1; 5 ne := ne-1; 6 if ne > 0 --> V(eu) 7 [] ne = 0 --> V(mu) 8 fi; 9 V(qu); a P(mu); nm := nm-1; b Critical Section c if nm > 0 --> V(mu) d [] nm = 0 --> V(eu) e fi |
Initially, ne = 0; nm = 0; eu = 1; mu = 0; qu = 1; Split binary semaphores: 0 ≤ eu + mu ≤ 1. eu satisfies UDWS, stated below. |
"A semaphore is called weak if absence of individual starvation among processes blocked at that semaphore cannot be guaranteed. For weak semaphores, one assumption is made, however, viz. (S1) a process that executes a V-operation on a semaphore will not be the one to perform the NEXT P-operation on that semaphore, if a process has been blocked at that semaphore. (S2) One of the waiting processes is allowed to pass that semaphore." We will call it UDWS.
The problem is that UDWS can be easily misinterpreted. Udding's assumption consists of two sentences as marked above.
Interpretation A: If we interpret S2 as "One of the waiting processes will be chosen whenever a V operation is executed and will be the next one to pass the semaphore" then S1 is redundant. Since the process to perform the next P operation has already been chosen from the waiting ones it is obviously not the process that just performed the V operation. This interpretation is equivalent to Morris' and I actually think this is what Udding meant. The fact that the first sentence is there may cause the reader to neglect the second sentence.
Interpretation B: We can also interpret the "is allowed to pass" of S2 as "waiting processes may pass (or permitted to pass) from now on, but they don't necessarily have to". Now S1 is obviously not redundant, since it imposes a restriction not implied by S2. Under this interpretation starvation can occur. This interpretation is not equivalent to Morris' assumption. This one looks much more appealing, but should no be chosen, because we can have the scenario described below.
ne=0, nm=0 eu=1 qu=0 mu=0 --------------1--------------2----------3--------------------a------ P1, P2, P3 at 1
Suppose P1 executes all of line 1. Further suppose, it also executes P(qu) in line 2. P1 cannot execute 3:P(eu) [because of UDWS] and therefore it waits at the beginning of line 3. We stll have P2, P3 waiting at 1:P(eu).
ne=1, nm=0 eu=1 qu=0 mu=0
--------------1--------------2----------3--------------------a------
P1 P1 P1
P2, P3
Let P2 execute line 1. It waits at the beginning of line 2 since qu = 0.
ne=2, nm=0 eu=1 qu=0 mu=0
--------------1--------------2--------3--------------------a------
P1 P1 P1
P2 P2
P3
ne=1, nm=1 eu=1 qu=1 mu=0
--------------1--------------2--------3--------------------a------
P1 P1 P1 P1
P2 P2
P3
ne=1, nm=1 eu=1 qu=0 mu=0
--------------1--------------2--------3--------------------a------
P1 P1 P1 P1
P2 P2 P2
P3
Let P2 continue through lines 4 and 5. When P2 arrives at line 6, ne = 0, so P2 executes 7:V(mu), and 9: V(qu). We now have: P1, P2 are waiting at a:P(mu). P3 is waiting at 1:P(eu). No process is waiting on qu.
ne=0, nm=2 eu=0 qu=1 mu=1
--------------1--------------2--------3--------------------a------
P1 P1 P1 P1
P2 P2 P2 P2
P3
Morris states his assumption (MDWS) for weak semaphores as follows: "If any process is waiting to complete a P(s) operation, and another process performs a V(s), the semaphore is not incremented, and one of the waiting processes completes its P(s)."
Execution of 7:V(mu) (see item 6 above) does not happen under MDWS. The moment P1 executes the 6:V(eu) (item 4 above) one of the processes waiting at 1:P(eu) completes it and increment ne.
In fact the execution would be different from way earlier. Strictly following MDWS, right after P1 has performed the 1:V(eu), one of the other processes has to perform the P(eu) operation. This will occur again when any other process finishes the line or executes a V(eu) operation, and if there are processes waiting at the beginning of line 1 they will sequentially execute P(eu). Starvation does not occur under Morris' definition.
Section 3.2 of Mark Ijbema is especially interesting in this discussion. It compares both assumptions and concludes that:
"This [Udding's assumption] is only marginally weaker [Morris' assumption], because the only difference is when the elected process makes the step into the guarded zone. In Udding's version this might be postponed until the elected process is scheduled again, instead of immediately. On first reading Udding's definition seems much weaker, because of the (redundant) first sentence. An interesting question is whether the algorithms would still be correct without the second sentence, that is, only demanding that the last process to execute the V operation cannot be the first to execute the P operation, if already processes were waiting. We look further into this question in appendix C."
The "(redundant)" was very important for me to understand the problem.
In Appendix C, we have the above situation described (for the second time). Based on this, Ijbema explains why buffered semaphores (defined by Ijbema) need to be used instead of polite semaphores (extensively explained in the thesis). Essentially polite semaphores only enforce one restriction: The process that executed the last V does not execute the next P. They only imply the first sentence of Udding's assumption. Buffered semaphores actually choose a new process when a V operation is executed from the waiting ones that will be the only one continuing its execution. The process "chosen" need not perform the P operation right away, but it is the only one to finish the P-operation. Buffered semaphores implement the second sentence of Udding's assumption under Interpretation A.
Udding is essentially right both in his solution and assumption, but the phrasing of the weak semaphore assumption should be improved.
This article was written by David Carral; edited by pmateti@wright.edu