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Consider the patterns HH, TH, HT , and TT for two tosses of a fair coin If you choose one of these patterns, I will choose another and then we toss a fair coin until one of these patterns occurs The winner is the person who chose the rstoccurring pattern For example, if you choose HH, I will choose TH If we see the sequence HTTTTH, then I win since the pattern TH occurred before the pattern HH My probability of beating you is 3/4! Here is why If the rst two tosses are HH, you win If the rst two tosses are TH, I win If the rst two tosses are HT then this can be followed by any number of T s, but eventually H will occur and I win If the rst two tosses are TT then this can be followed by any number of T s but eventually H will occur and I will win If you are to win, you must toss HH on the rst two tosses; this is the only way you can win and it has probability 1/4 If a T is tossed at any time, I will win, so my probability of winning is 3/4 The fact that the patterns HH, TH, HT , and TT are equally likely for a fair coin may be observed by a game player who may think that any choice is equally good is irrelevant; the choice of pattern is crucial, as is the fact that he is allowed to make the rst choice If you choose TT , then the only way you can win is by tossing two tails on the rst two tosses I will choose HT and I will win 3/4 of the time A sensible choice for you is either TH or HT, and then my probability of beating you is only 1/2 If we consider patterns with three tosses, as shown in the table above, my minimum probability of beating you is 2/3! (And if you do not choose well, I can increase that probability to 3/4 or 7/8!) This can be tried by simulation It is puzzling to note that no matter what pattern you choose, I will probably beat you This means that if we play the game twice, and I win in the rst game, then you can choose the pattern I chose on the rst game, and I can still probably beat you Probabilities then are not transitive, so if pattern A beats pattern B and pattern B beats pattern C, then it does not follow that pattern A will necessarily beat pattern C.

i n t e r n a l s t a t i c void E x it < T > ( pa rams T [ ] loc k s ) { Array Sort ( lo c k s ) j I I E x it t h e loc k s i n reve r s e sorted orde r for ( i nt i loc k s Length - 1 j i > = a j i - - ) Monitor E xit ( loc k s [ i ] ) j

This code has some disadvantages One clear disadvantage is the performance overhead for doing a sort We also have to do it twice, once for

the same locks array to both methods, we could sort it in place in E nt e r and then skip the sort entirely inside of Exit assuming the same array is supplied Another disadvantage is that locks themselves don't always have unique keys associated with them When coding in C++ with C R ITICAL_S E CTIONs, you can sort on the memory address; and with kernel objects, you can use the HAND L E value Both are guaranteed unique and stable But CLR monitors can be any kind of CLR object, so you need to implement ordering at some higher level (hence the restriction in Mu lt i Lo c k H e l pe r < T > above that the generic argument T implements ICompa r a b l e < T We could do this in our Ba n kAc c o u n t example by combining the m_i d and m_sy n c Loc k fields into a single comparable object

006% 10% 074% 61% 0% 70% 0% 10%

This apparent paradox, that probabilities are not transitive, continues with patterns involving patterns of length 3 We showed above the average waiting times for each of the eight patterns that can occur when a fair coin is tossed (Table 76) Now let us play the coin game again where A is the rst player and B is the second player We show the probabilities that B beats A in Table 77 Note that letting > means beats (probably) TTH > HTT > HHT > THH > TTH!

Lock Leveling All of this talk about ordering locks during acquisition brings us to our next technique for avoiding deadlocks: lock leveling This technique is commonly known under several other guises: lock ranking, lock hierarchies, and lock ordering, among others We already stated that if threads always acquire locks in a consistent order, there will be no dead locks, but it may not be obvious why this is true Cycles are required to pro duce a deadlock, and consistent ordering (with no exceptions) eliminates the possibility of cycles Imagine we assign a unique level to each lock in the system (This is stronger than the previous example, where only like locks needed to be disambiguated ) Then, if a thread only waits for locks with a level "less" than the lowest level already currently held, it is enough to guarantee dead lock freedom by construction Strict adherence to the leveling scheme can be statically verified in the best case, and dynamically verified in the worst