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roulette the wheel has 37 compartments numbered 0, 1, , 36, where the odd numbers are black and the even numbers except for the zero are red An interesting question that naturally arises is: what is the probability that during the next m spins of the wheel there will be some sequence of r consecutive spins that all result either in r black numbers or in r red numbers for a given value of r This question can be answered by Markov chain theory The idea is to de ne a Markov chain with r + 1 states including an absorbing state The process is said to be in state 0 when the last spin of the wheel resulted in a zero, while the process is said to be in state i with 1 i < r when the same colour (red or black) appeared in the last i spins but this colour did not appear in the spin preceding the last i spins The process is said to be in state r when the last r spins of the wheel have resulted in the same colour The state r is taken as an absorbing state; imagine that the wheel sticks to the colour of the success run once a success run of length r has occurred A success run of length r is said to occur when state r is reached Denote by Xn the state of the process after the nth spin of the wheel, with X0 = 0 by convention The stochastic process {Xn } is a discrete-time Markov chain Its one-step transition probabilities are given by 1 36 , p01 = , 37 37 18 1 , pi0 = pi,i+1 = pi1 = 37 37 prr = 1 p00 =

The other pij are zero Since state r is absorbing, it is not possible that the process has visited state r before time t when the process is in some state i = r at time t Hence P {more than m spins are needed to get a success run of length r} = P {Xk = r for k = 1, , m | X0 = 0} = P {Xm = r | X0 = 0} = 1 P {Xm = r | X0 = 0}

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transfer of CAP components (operations and errors) through an open TCAP dialogue; TC End is used to close a TCAP dialogue GPRS Reference Number The TCAP dialogue between two TC-Users, such as gprsSSF and gsmSCF, take care that TCAP messages exchanged between SGSN and SCP are delivered to the correct application in this SGSN or SCP In the example in Figure 549, the SGSN has several gprsSSF instances active, as well as a number of MAP applications, eg with the HLR The SCP has several gsmSCF instances active and also a number of MAP applications, eg for any time GPRS interrogation (see 6) TCAP uses the TCAP transaction identi er to couple an incoming TCAP message to an application When a new TCAP dialogue is established, a new application instance is invoked When TCAP segmentation is applied, as for CAMEL for GPRS control, the TCAP dialogue between the gprsSSF and the gsmSCF may be released while the gprsSSF and gsmSCF themselves remain active An incoming TC Begin message during an active CAMEL dialogue could therefore not be offered to the correct application Instead, a TC Begin from the gprsSSF, carrying the Event Report GPRS (PDP Context Establishment Acknowledgement), would start a new gsmSCF application instance To overcome the above, a GPRS reference number is used This information element has the following structure (see 3GPP TS 29078 [106]):

The desired probability that a success run of length r will occur during the rst m (m) spins of the wheel is thus p0r How can we calculate this probability for r = 26 when N is of order 8 million (a rough estimate for the number of spins of the roulette wheel in Monte Carlo between the date of the founding of the casino and the date of 18 August 1913) It is not advised to multiply the 27 27 matrix P = (pij ) 8 million times by itself A more clever computation is based on P2 = P P, P4 = P2 P2 , P8 = P4 P4 , etc Taking k = 23, we have 2k is about 8 million Hence it suf ces to do 23 matrix (m) multiplications to get p0,26 for m = 223 This gives the probability 0061 Another

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