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The m-procedures mgd and jmgd

The next example shows a fundamental limitation of the gend procedure. The values for the individual demands are not limited to integers, and there are considerablegaps between the values. In this case, we need to implement the moment generating function M D rather than the generating function g D .

In the generating function case, it is as easy to develop the joint distribution for { N , D } as to develop the marginal distribution for D . For the moment generating function, the joint distribution requires considerably more computation.As a consequence, we find it convenient to have two m-procedures: mgd for the marginal distribution and jmgd for the joint distribution.

Instead of the convolution procedure used in gend to determine the distribution for the sums of the individual demands, the m-procedure mgd utilizes them-function mgsum to obtain these distributions. The distributions for the various sums are concatenated into two row vectors, to which csort is applied to obtain the distributionfor the compound demand. The procedure requires as input the generating function for N and the actual distribution, Y and P Y , for the individual demands. For g N , it is necessary to treat the coefficients as in gend. However, the actual values and probabilitiesin the distribution for Y are put into a pair of row matrices. If Y is integer valued, there are no zeros in the probability matrix for missing values.

Noninteger values

A service shop has three standard charges for a certain class of warranty services it performs: $10, $12.50, and $15. The number of jobs received in a normal workday can be considered a random variable N which takes on values 0, 1, 2, 3, 4 with equal probabilities 0.2. The job types for arrivals may be represented by an iidclass { Y i : 1 i 4 } , independent of the arrival process. The Y i take on values 10, 12.5, 15 with respective probabilities 0.5, 0.3, 0.2. Let C be the total amount of services rendered in a day. Determine the distribution for C .

SOLUTION

gN = 0.2*[1 1 1 1 1];         % Enter dataY = [10 12.5 15];PY = 0.1*[5 3 2];mgd                           % Call for procedure Enter gen fn COEFFICIENTS for gN  gNEnter VALUES for Y  Y Enter PROBABILITIES for Y  PYValues are in row matrix D; probabilities are in PD. To view the distribution, call for mD.disp(mD)                      % Optional display of distribution          0    0.2000   10.0000    0.1000    12.5000    0.0600   15.0000    0.0400    20.0000    0.0500   22.5000    0.0600    25.0000    0.0580   27.5000    0.0240    30.0000    0.0330   32.5000    0.0450    35.0000    0.0570   37.5000    0.0414    40.0000    0.0353   42.5000    0.0372    45.0000    0.0486   47.5000    0.0468    50.0000    0.0352   52.5000    0.0187    55.0000    0.0075   57.5000    0.0019    60.0000    0.0003
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We next recalculate [link] , above, using mgd rather than gend.

Recalculation of [link]

In [link] , we have

g N ( s ) = 1 5 ( 1 + s + s 2 + s 3 + s 4 ) g Y ( s ) = 0 . 1 ( 5 s + 3 s 2 + 2 s 3 )

This means that the distribution for Y is Y = [ 1 2 3 ] and P Y = 0 . 1 * [ 5 3 2 ] .

We use the same expression for g N as in [link] .

gN = 0.2*ones(1,5); Y = 1:3;PY = 0.1*[5 3 2];mgd Enter gen fn COEFFICIENTS for gN  gNEnter VALUES for Y  Y Enter PROBABILITIES for Y  PYValues are in row matrix D; probabilities are in PD. To view the distribution, call for mD.disp(mD)          0    0.2000    1.0000    0.1000     2.0000    0.1100    3.0000    0.1250     4.0000    0.1155    5.0000    0.1110     6.0000    0.0964    7.0000    0.0696     8.0000    0.0424    9.0000    0.0203    10.0000    0.0075   11.0000    0.0019    12.0000    0.0003P3 = (D==3)*PD' P3 =   0.1250ED = D*PD' ED =   3.4000P_4_12 = ((D>=4)&(D<=12))*PD' P_4_12 =  0.4650P7 = (D>=7)*PD' P7 =   0.1421

As expected, the results are the same as those obtained with gend.

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If it is desired to obtain the joint distribution for { N , D } , we use a modification of mgd called jmgd . The complications come in placing the probabilities in the P matrix in the desired positions. This requires some calculations to determine the appropriate size of the matrices used as well as a procedure to put each probability in theposition corresponding to its D value. Actual operation is quite similar to the operation of mgd, and requires the same data format.

A principle use of the joint distribution is to demonstrate features of the model, such as E [ D | N = n ] = n E [ Y ] , etc. This, of course, is utilized in obtaining the expressions for M D ( s ) in terms of g N ( s ) and M Y ( s ) . This result guides the development of the computational procedures, but these do notdepend upon this result. However, it is usually helpful to demonstrate the validity of the assumptions in typical examples.

Remark . In general, if the use of gend is appropriate, it is faster and more efficient than mgd (or jmgd). And it will handle somewhat larger problems. But both m-procedures workquite well for problems of moderate size, and are convenient tools for solving various “compound demand” type problems.

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Source:  OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6
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