Ross, S. M. (1970). Applied Probability Models with Optimization Applications. San Francisco: Holden-Day.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is used, is the initial condition, and the minimum is with respect to all policies. The difference is considered, where is some distinguished state, and one then seeks conditions under which converges as to a solution to the average cost optimality equations, thereby giving an optimal policy [41]. To make this approach work, in [42] 43] and other papers it is assumed that there is a finite valued function on the state space and a constant such that (1) for all states and all sufficiently close to unity. This condition appears to be far removed from the initial problem statement. ....

....Dynkin s formula gives an upper bound on where we have used the inequality , which follows from (20) Thus, for all and we have (21) where . The proof of P2) follows on letting (22) It is well known that the average cost optimal control problem is plagued with counterexamples [1] 11] 39] [41]. It is of some interest then to see why Theorem 3.1 does not fall into any of these traps. Consider first [41, p. 142, counterexamples 1 and 2] In each of these examples the process, for any policy, is completely nonirreducible in the sense that for all times and all policies . It is clear then ....

S. M. Ross, "Applied probability models with optimization applications, " in Dover Books on Advanced Mathematics, 1992.

....stationary policies that either always keep the server on or are regenerative. This step is presented in Section 5. We model the problem as a semi Markov decision process with an uncountable state space [0; 1) and unbounded cost functions. The available theory of semi Markov decision processes [20, 22] does not cover such models. The theory of in nite state space Markov decision processes (MDP) with unbounded cost functions was developed during the last decade, see Sennott [24, 25] Sch al [23] provided sucient conditions for the existence of stationary optimal policies in uncountable ....

Ross, S. M. (1970). Applied Probability Models with Optimization Applications. Holden-Day, San Francisco.

....that B ffi 0 is approachable by DM2) By following the same arguments we see that B is excludable by DM2 if B KL 6= Thus, if we prove (i) part (b) of the Theorem follows. Step 3: Here we prove (i) and (ii) Then by step 1 and step 2 the Theorem follows. To this end, note that [see [4] Th. 7.5] R(f; g) lim P n Gamma1 P n Gamma1 j = j : 14) Denote N : m) and CN : m=0 Cm) Let f be any stationary policy for DM1. The policy f is well defined by the probability to choose the fast service mode when the queue is empty. ....

S. M. Ross, Applied Probability models with Optimization Application, Holden-Day, San Francisco, 1970.

....all the failure times prior to the failure, # 12 . nni SXXXX (2.44) If the random variables . 12 , n XXXare independent and identically distributed, all with pdf s of ( fx, the random process described by these variables is referred to as an Ordinary Renewal Process [Cox62] Ross70] The details of the Renewal Process are shown in Appendix E. Given the random process described by Eq. 2.44) the distribution function of n S is provided by convolving each individual distribution function ( Ft. The convolution of two functions is defined as [Brac65] Papo65] ....

....the distribution function, giving, htHt = 2.56) Using Eq. 2.50) to evaluate the derivative results in, htft (2.57) and using Eq. 2.54) as a substitute for the right side of Eq. 2.57) results in, htfthtxfxdx= 2.58) Eq. 2. 58) is known as the Renewal Equation [Ross70] To solve the renewal equation, the Laplace transform will be used. The transform of the probability density function is, fsefxdx (2.59) and the transform of the renewal function is, hsehxdx (2.60) Using the convolution property of the Laplace transform [Brac65] an equation ....

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Ross S., Applied Probability Models with Optimization Applications, Holden-- Day, 1970.

.... The results of our analysis have an immediate practical application as the optimal policy can easily be computed once basic statistic parameters defining the tra#c of requests are known, by solving an optimization problem by means of very commonly used methods of operations research [8] 10] [14], 15] II. A Multi class Semi Markov Decision Model for CAC We consider a decision model that allows for multiple class calls that di#er in their QoS requirements and tra#c parameters, allowing for call transitions among classes. Each cell in the system has to give service to N classes of call ....

S. Ross, Applied probability models with optimization applications, Holden-Day, 1970.

....of state x. Define oe : minf n : n 1 and x( n ) 0g, the first entrance time to state 0. From the reciprocal relationship between steady state probabilities and mean recurrence times, we have E[oe j x( 0 ) 0] 0) Note also that from the renewal relationship (see Theorem 7. 5 of Ross [19]) x (x)jxj = 6 jx(n)jjx( 0 ) 0] E[oejx( 0 ) 0] From the one step dynamic programming recursion, the numerator above satisfies E[ n=0 jx( n )j j x( 0 ) 0] y p 0y V (y) Hence x (x)jxj = 0) y p 0y V (y) Since the LHS is finite, so is the sum on the RHS. Hence, we see that ....

S. M. Ross, Applied Probability Models with Optimization Applications. San Francisco: Holden--Day, 1970. 33

....optimistic. We consider the applicability of the Markov renewal model in the operational phase next. During the operational phase no changes are made to the software, that is, the sequence of runs can be described by the homogeneous DTMC. The well developed theory of Markov renewal processes [5] [21] and its limiting results can be used to derive a number of measures other than the distribution of the time between failures, such as ffl the expected number of failures MF (t) E[NF (t) and the expected number of successes MS (t) E[NS (t) in the interval (0; t] ffl the probability of ....

S. M. Ross. Applied Probability Models with Optimization Applications. Holden -- Day, 1970.

....of independence is overly optimistic. Consider the applicability of the MRP approach in the operational phase. During the operational phase, assume that no changes are made to the software. Then the sequence of runs can be described by the homogeneous DTMC. The well developed theory of MRP [4] [26] and its limiting results can be used to derive several measures other than the distribution of the time between failures, such as the: mean number of failures and mean number of successes in the interval ; probability of success at time : instantaneous availability; limiting probability ....

S. M. Ross, Applied Probability Models with Optimization Applications: Holden-Day, 1970.

....an analytically simpler approximation, whereby the number of slots in each subfield is a random variable, with the expected values of each equal to the actual fixed number of slots, and , respectively. Then, we model the two subfields as the states of an alternating renewal process [11]. Subfield 1 of the data segment is the first state of the alternating renewal process, and the Subfield 2 is the second state. The probability that the next slot belongs to Subfield 1 is C B (the average fraction of the time spent in state 1) and the probability that it belongs to Subfield 2 ....

S.M.Ross, Applied Probability Models with Optimization Applications, Dover Publications Inc., NY, 1992.

....s) is incurred, one for each term w ls B l Gamma w sl B s , where 1 l; s M and l 6= s. ffl The next state is determined according to the transition rate ff u (n;p) given in Eq. 5. 7) From the above discussion it follows that we have a constrained continuous time MDP problem to solve (see [53, 54, 55]) We will now show that we can search for the optimal policy over a smaller set instead of the original set F 0 . First, note that the set F 0 has control policies that depend on the time at which the admission control decision is made. For example, it includes the control policy that ....

....develop a pricing mechanism that prevents such behavior by users. 6.2 MDP Formulation of Resource Allocation In this section, we formulate the resource allocation problem as an MDP. Efficient methods exist to solve such problems, namely value iteration, policy iteration, and linear programming [53, 57, 58]. In the interest of space, we merely formulate the problem as an MDP and refer the reader to the references cited above for solution techniques. We first describe the problem scenario. We consider a single link with a finite capacity that supports a variety of applications. We assume that ....

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Sheldon M. Ross. Applied Probability Models with Optimization Applications. Holden-Day, 1970.

....tools necessary to formulate the dynamic programming problem for SmartPath. There is a vast literature dealing with the theory and application of dynamic programming. Bellman s book [51] is an excellent reading. For applications of dynamic programming see [50] Most of the above results are from [52] and [53] 98 5.2.2 Dynamic Load Balancing: Problem Formulation For the distributed SmartPath simulation let X j i , the state of a processor j at time i, be the number of vehicles simulated in that processor, and let the state of the system be X i = kX j i k 1 = max j X j i Assuming ....

S. M. Ross, Applied probability models with optimization applications. Holden-Day, 1970.

.... A useful property of the above model which proves to be convenient for theoretical analysis is that the size of the network converges rapidly to a value very close to = henceforth we denote this quantity by the parameter N ) This is because the model behaves as an in nite server Poisson queue ([7]) Before we state our results we need a little terminology. Let G t be the network at time t (G 0 has no vertices) We analyze the evolution in time of the stochastic process G = G t ) t 0 . We state only the main results and their intuitive justi cations; for rigorous proofs we refer to [5] ....

S. Ross. Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970. 5

....costs over an unbounded horizon: L(OE; k) lim n 1 C(n; OE; k) n = P 1 i=1 c i (OE; k)p i (OE; k) P 1 i=1 ip i (OE; k) 8) where C(n; k) are the expected costs in time interval (0; n Delta] Eq. 8) is a well known result from renewal reward theory (see e.g. Ross [9]) Second, the expected discounted costs over an unbounded horizon are determined by summing the expected discounted values of the costs over an unbounded horizon, 7 failure resistance [number of stones] s breaches of armour layer preventive repair r o unit time 0 I I I I F X Figure 2: ....

Sheldon M. Ross. Applied probability models with optimization applications. Holden-Day, 1970.

....process the arrival time of a random element is uniform in [0; t] p(t) 1 t Z t 0 e (t ) N d = 1 t N(1 e t=N ) Our process is similar to an infinite server Poisson queue. Thus, the number of nodes in the graph at time t has a Poisson distribution with expectation tp(t) see [10],pages 18 19) For t = N) E[jV t j] N) When t=N 1, E[jV t j] N o(N) We can now use a tail bound for the Poisson distribution [1] page 239] to show that for t = N) P r jjV t j E[jV t j]j p bN log N 1 1=N c for some c 1. 2 The above theorem assumed that the ....

S.M. Ross. Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970. 8

....process the arrival time of a random element is uniform in ##;t#) p#t## # t # # # e ######## d # # t N## # e #### #: Our process is similar to an infinite server Poisson queue. Thus, the number of nodes in the graph at time t has a Poisson distribution with expectation tp#t# (see [10][pages 18 19] For t ### N#, E##V # ## # ##N#.Whent=N # #, E##V # ###N # o#N#. We can now use a tail bound for the Poisson distribution [1] page 239] to show that for t## #N#, Pr # ##V # ##E##V # #### # bN ### N # # # # #=N # for some c #. # The above theorem assumed ....

S.M. Ross. Applied Probability Models with Optimization Applications, Holden-Day, San Francisco, 1970.

....t=0 fl t R(x t ; t (x t ) g) Then for any n = 0; 1; the value iteration function v n satisfies kv n Gamma V 1 k fl n kv 0 Gamma V 1 k. That is, V I is a contraction mapping and successive application of V I will make v 0 converge to V by Banach fixed point theorem (see Ross [145] or Hern andez Lerma s book [73] on the discussion of this theorem) 2.1.1.3 Infinite horizon average reward It turns out that for this criterion, it is much more difficult to establish the theories about the existence of an optimal policy and analyze. Even though the theories for finite ....

S. M. Ross, Applied Probability Models with Optimization Applications. Dover Books on Mathematics, 1992.

....processes of which Q learning is a special case. We then discuss TD( and show that it is also a special case of our theorem. Markov decision problems A useful mathematical model of temporal credit assignment problems, studied in stochastic control theory (Aoki, 1967) and operations research (Ross, 1970), is the Markov decision problem. Markov decision problems are built on the formalism of controlled Markov chains. Let S = 1; 2; N be a discrete state space and let U(i) be the discrete set of actions available to the learner when the chain is in state i. The probability of making a ....

Ross, S. M. (1970). Applied Probability Models with Optimization Applications. San Francisco: Holden-Day.

.... if V (i; j) V (i; j) 8 i; j : 0 i B; 0 j: If f is a stationary policy which chooses actions according to f(i; j) arg min a ae C(i; j; a) B Gamma1 X k=0 1 X l=0 P i;j;k;l (a)V (k; l) oe ; 0 i B; 0 j; then V f (i; j) V (i; j) 0 i B; 0 j and hence f is optimal [12] (arg min a fF (a)g denotes the value of a where F (a) is minimal) Substituting the transition probabilities, we can write the above Equation in a simpler form: f(i; j) arg min a ae C(i; j; a) B Gamma1 X k=0 P i;j;k;j 1 (a)V (k; l) oe ; 0 i B; 0 j: 1) Thus we have formulated ....

....and the look ahead n minimal cost functions. Therefore, when the cost C tends to zero with time, the look ahead cost function series V n converges to the minimal cost function V . We also derive results on the speed of convergence. The proofs of the above statements follow the approach given in [12]. Let V 0 (i; j) C(i; j; stop) 0 i B; 0 j; and for n 0, V n (i; j) min ae C(i; j; stop) B Gamma1 X k=0 P i;j;k;j 1 (continue)V n Gamma1 (k; l) oe 0 i B; 0 j: 2) If we start in state (i; j) V n (i; j) is the minimal expected cost if the process can go at most n stages ....

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S. M. Ross, Applied Probability Models with Optimization Applications. Dover Publications, Inc., New York, 1992. 19

....even for simple models, hence an accurate computer aided analysis of possible decisions is needed to obtain decision rules that lead to a sort of optimality. Previous results on decisions in Discrete Time Markov Chains (DTMC) Continuous Time Markov Chains (CTMC) and Semi Markov Processes (SMP) [8, 4] provide the analysis of the infinite time horizon problem with and without discounting. The case of finite time horizon was analyzed for DTMCs [8, 4] and for CTMCs [7] The analysis of decision processes with memoryless (Markov) property at the decision instances is based on the reward analysis ....

....Previous results on decisions in Discrete Time Markov Chains (DTMC) Continuous Time Markov Chains (CTMC) and Semi Markov Processes (SMP) 8, 4] provide the analysis of the infinite time horizon problem with and without discounting. The case of finite time horizon was analyzed for DTMCs [8, 4], and for CTMCs [7] The analysis of decision processes with memoryless (Markov) property at the decision instances is based on the reward analysis of the subprocesses, referred to as subordinated processes, between the consecutive decision instances. It turned out that the optimal decision in an ....

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S. M. Ross. Applied Probability Models with Optimization Applications. Dover Publications, Inc., New York, 1992.

....there exists 0 1 for which if 1 0 then for the systems that di er only in the discount rate, the same f policy will be optimal. As a consequence the optimal policy for the case of no discount can be determined as the limiting policy for systems with discount rate, if tends to 1 [5]. For this we should nd the optimal policy for systems where discount rate is involved in the cost structure. We have the methods of policy iteration and successive approximation. 1. policy iteration For an initial f policy we calculate the expected total cost function (assumed discount ....

S. M. Ross. Applied Probability Models with Optimization Applications. Dover Publications, Inc., New York, 1992.

.... f (i; j) V (i; j) for all i; j : 0 i B; 0 j : If f is a stationary policy which chooses action according to f(i; j) arg min a ae C(i; j; a) B Gamma1 X k=0 1 X l=0 P i;j;k;l (a)V (k; l) oe ; 0 i B; 0 j (1) then V f (i; j) V (i; j) 0 i B; 0 j hence f is optimal (Ross, 1992) (arg min a fF (a)g denotes a value of a where F (a) is minimal) Thus we have formulated the problem as a Markov Decision Process, for which stationary optimal policy exists, and it is determined by Equation 1, Ross, 1992) Substituting the transition probabilities we can write Equation 1 into ....

.... 0 j (1) then V f (i; j) V (i; j) 0 i B; 0 j hence f is optimal (Ross, 1992) arg min a fF (a)g denotes a value of a where F (a) is minimal) Thus we have formulated the problem as a Markov Decision Process, for which stationary optimal policy exists, and it is determined by Equation 1, (Ross, 1992). Substituting the transition probabilities we can write Equation 1 into a simpler form: f(i; j) arg min a ae C(i; j; a) B Gamma1 X k=0 P i;j;k;j 1 (a)V (k; l) oe ; 0 i B; 0 j : pappfen.tex; 5 06 1996; 14:22; no v. p.5 6 Andr as Pfening and Mikl os Telek The next step is to ....

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S. M. Ross. Applied Probability Models with Optimization Applications. Dover Publications, Inc., New York, 1992.

.... V (i; j) V (i; j) 8 i; j : 0 i B; 0 j: If f is a stationary policy which chooses actions according to (for 0 i 5 B; 0 j ) f(i; j) arg min a ae C(i; j; a) B Gamma1 X k=0 P i;j;k;j 1 (a)V (k; j 1) oe ; 1) then V f (i; j) V (i; j) 0 i B; 0 j and hence f is optimal [11] (arg min a fF (a)g denotes the value of a where F (a) is minimal) Thus we have formulated the problem as a Markov Decision Process, for which a stationary optimal policy exists and is determined by Equation 1. The next step is to derive V (i; j) 8(i; j) the minimal expected cost when the ....

....show that the cost C is the upper bound on the difference of the optimal and the look ahead n cost functions. Therefore when C tends to zero with time, the look ahead cost function series V n converges to the minimal cost function V . The proofs of the above statements follow the approach given in [11]. Let V 0 (i; j) C(i; j; rej) 0 i B; 0 j; and for n 0; 0 i B; 0 j, V n (i; j) min ae C(i; j; rej) B Gamma1 X k=0 P i;j;k;j 1 (cont)V n Gamma1 (k; j 1) oe : 2) If the system starts in state (i; j) V n (i; j) is the minimal expected cost if the process can go at most n ....

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S. M. Ross, Applied Probability Models with Optimization Applications. Dover Publications, Inc., New York, 1992.

....of an optimal constrained pair of stationary policy and stopping time utilizing a Lagrange multiplier approach. The proof is executed by applying a Lagrange multiplier method developed by Frid [6] Beutler and Ross [3] and Sennott [18] Also, using the idea of the one step look ahead (OLA cf.[16]) policy an optimal constrained pair is derived concretely. The constrained Markov deteriorating system is illustrated as an example. In the reminder of this section, we shall give the problem formulation referring to Hordijk [8] Also, an optimal constrained pair of policy and stopping time is ....

S. M. Ross. Appliedprobability Models with Optimization Applications. Holden-Day,1970.

....threats against existence and external assaults. 2 Model Formulation Suppose disastrous threats arrive according to independent Poisson processes. Let S denote the expected loss to the principal if a disaster occurs. Then the disaster process can be modeled as a compound Poisson process (see e.g. [5] p. 23) Let h denote the arrival rate of disaster events apart from the agent s e#ort; h is called the natural hazard rate below. The e#ort of an alert and circumspective agent reduces the rate h. The agent s e#ort level u is measured as the proportion of arriving disasters e#ectively obviated. ....

Ross, S.M. (1971), Applied Probability Models with Optimization Applications.

.... Markov chain is a stochastic process fXm ; m = 1; 2; 3; g with a nite or countable state space adhering to the following property: PfXm 1 = j j Xm = i; Xm 1 = i m 1 ; X 2 = i 2 ; X 1 = i 1 g = PfXm 1 = j j Xm = ig; 1) for all states i 1 ; i 2 ; i m 1 ; i; j, and all m 1 (Ross, 1970). In other words, the probability that the next state Xm 1 is j, given the current state (Xm = i) and any past state (X 1 = i 1 ; Xm 1 = i m 1 ) is dependent only upon the current state i. In general, a stochastic process that satis es Equation 1 is said to be rst order Markovian. If ....

....process (SMP) is a generalization of a Markov chain allowing arbitrary state durations. We let Q ij (t) be the probability of remaining in state i for time t before transitioning to state j. If we let P ij = Q ij (1) the P ij de ne the transition probabilities of the embedded Markov chain (Ross, 1970) and it follows that P j P ij = 1. We let F ij (t) Q ij (t) P ij be the conditional probability of remaining in state i for time t given the system has just entered state i and will transition to state j. Ross (1970) provides further details on Markov chains and semi Markov processes. An ....

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Ross, S. M.: 1970, Applied Probability Models with Optimization Applications. New York: Dover Publications, Inc.

....of the process beyond T is a probabilistic replica of the whole process starting at zero. Define the limiting probability P j as: X(t) j , Referring to the time between two regeneration points as a cycle, the limiting probability P j can be computed as given in the following theorem [4]. Theorem 1 If T has an absolutely continuous component (that is, it has a density on some interval) and E[T] then, P j = E[Amount of time in state j during one cycle] E[time of one cycle] 1) for all . Definition 2: If a stochastic process which makes transitions from state to state in ....

....to state in accordance with a Markov chain, and if the process is also a regenerative process, the stochastic process is called a Markov regenerative process. t 0 , P j P t lim = j E j 0 6 For Markov regenerative processes, the limiting probability P j can be computed as follows[4]. Theorem 2 Let P(j) be the steady state probability for state j (P(j) equals the (long run) proportion of transitions which are into state j) D j be the mean time spent in state j per transition, and P j be the limiting probability (P j equals the (long run) probability that the process is in ....

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S. M. Ross, Applied Probability Models with Optimization Application, Holden-day, San Francisco, 1970.

....is a continuous time dynamic system consisting of a countable state set, X , and a finite action set, A. Suppose that the system is originally observed to be in state x 2 X , and that action Accepted at the NIPS 94 conference, Nov. 1994, Denver, CO 2 a 2 A is applied. A semi Markov process [9] then evolves as follows: ffl The next state, y, is chosen according to the transition probabilities P xy (a) ffl A reward rate ae(x; a) is defined until the next transition occurs ffl Conditional on the event that the next state is y, the time until the transition from x to y occurs has ....

S. M. Ross. Applied Probability Models with Optimization Applications. Holden-Day, San Francisco, 1970.

....of the process beyond T is a probabilistic replica of the whole process starting at zero. Define the limiting probability P j as: X(t) j , Referring to the time between two regeneration points as a cycle, the limiting probability P j can be computed as given in the following theorem [13]. Theorem 1 If T has an absolutely continuous component (that is, it has a density on some interval) and E[T] then, P j = E[Amount of time in state j during one cycle] E[time of one cycle] for all . Definition 2: t 0 , P j P t lim = j E j 0 5 If a stochastic process which ....

S. M. Ross, Applied Probability Models with Optimization Application, Holden-day, San Francisco, 1970. 21 x y The hearing region of y Fig. .1 Illustration of the hidden terminal problem in a multihop network. z The hearing region of x Fig.2 The Markov chain for the channel CH(x)

....provided that Conditions 1 and 2 in [6, p. 391] are satisfied. 2 A Semi Markov Decision Problem Solving directly for problem P1 is a difficult task, since the structure of the cost criterion (1. 1) does not fit the standard Semi Markov Decision Process (SMDP) setting (cf. Lippman [14] Ross 3 [16]) To see that, let us rewrite V (x; u) as, cf. 1.1) V (x; u) lim t 1 1 t E u x 2 4 X 0tn t Z t n 1 t n X( d fl 1(A n = s) Gamma Z t N(t) 1 t X( d 3 5 ; x 2 IN; 2:1) where the symbol E u x stands for the expectation operator given that X(0) x and that policy u is ....

....under a threshold policy u l and define S x : infft n ; n 2 : X(t n ) xg; 4:4) given X(0) x, x 2 IN. Clearly, for every x 2 IN, the process fX(t) t 0g is a regenerative process with regeneration point S x . If E u l x [S x ] 1 for every x 2 IN, then (4.3) follows from Theorem 7. 5 in [16]. Let us show that E u l x [S x ] 1 for every x 2 IN. It is easily seen that X : fX n g n1 is an irreducible, aperiodic Markov chain. Let us show that all the states of this chain are recurrent non null. Clearly, under policy u l X n 1 = U n X n 1(X n l) U n l; 4.5) for all n 1, ....

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S. M. Ross, Applied Probability Models with Optimization Applications , Holden-Day, San Francisco, 1970.

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Ross, S.M., Applied Probability Models with Optimization Applications. Holden-Day Inc., San Francisco, 1970.

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S. M. Ross, Applied probability models with optimization applications. Holden-Day, 1970.

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Ross, S.M., Applied Probability Models with Optimization Applications, HoldenDay, San Francisco, 1970.

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S.M. Ross. Applied Probability Models with Optimization Applications. Holden-Day, San Francisco, California, 1970.

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Ross, S. M. (1971) Applied probability models with optimization applications. HoldenDay.

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S.M. Ross, Applied Probability Models with Optimization Applications, San Fransisco: Holden-Day, 1970.

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S.M. Ross. Applied probability models with optimization applications. Holden-Day, San Francisco [etc.], 1970.

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