j = 1, , K in VS .NET

Generate Denso QR Bar Code in VS .NET j = 1, , K
j = 1, , K
Recognizing Quick Response Code In .NET
Using Barcode Control SDK for Visual Studio .NET Control to generate, create, read, scan barcode image in .NET applications.
(5611)
Encode QR-Code In .NET
Using Barcode creator for .NET Control to generate, create QR Code JIS X 0510 image in VS .NET applications.
K in conjunction with the normalizing equation j =1 j = 1 The relative visit frequencies to the stations are proportional to these equilibrium probabilities To see this, let
Quick Response Code Reader In .NET Framework
Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.
j = the long-run average arrival rate of customers at station j Since i is also the rate at which customers depart from station i, we have that i pij is the rate at which customers arrive at station j from station i This gives the traf c equations
Barcode Creator In .NET
Using Barcode creator for VS .NET Control to generate, create barcode image in .NET framework applications.
j =
Bar Code Decoder In .NET
Using Barcode scanner for .NET framework Control to read, scan read, scan image in VS .NET applications.
i pij ,
Making Denso QR Bar Code In C#.NET
Using Barcode creator for VS .NET Control to generate, create QR Code JIS X 0510 image in .NET framework applications.
j = 1, , K
Generate QR Code ISO/IEC18004 In .NET
Using Barcode printer for ASP.NET Control to generate, create QR image in ASP.NET applications.
(5612)
Generate QR Code JIS X 0510 In VB.NET
Using Barcode drawer for .NET framework Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications.
The solution of the equilibrium equations (5611) of the Markov matrix P is unique up to a multiplicative constant Hence, for some constant > 0, j = j , j = 1, , K (5613)
Encoding Bar Code In .NET
Using Barcode encoder for Visual Studio .NET Control to generate, create bar code image in .NET framework applications.
Denote by Xj (t) the number of customers present at station j at time t The process {(X1 (t), , XK (t))} is a continuous-time Markov chain with the nite state space I = {(n1 , , nK ) | ni 0, K ni = M} i=1 Theorem 562 The equilibrium distribution of the continuous-time Markov chain {X(t) = (X1 (t), , XK (t))} is given by
ECC200 Creation In VS .NET
Using Barcode generation for .NET framework Control to generate, create DataMatrix image in VS .NET applications.
p(n1 , , nK ) = C
Create Barcode In .NET
Using Barcode creator for .NET framework Control to generate, create bar code image in VS .NET applications.
k k
Leitcode Creator In Visual Studio .NET
Using Barcode generator for VS .NET Control to generate, create Leitcode image in .NET framework applications.
(5614)
Bar Code Generation In Visual C#
Using Barcode maker for .NET framework Control to generate, create barcode image in Visual Studio .NET applications.
for some constant C > 0
Making Code 39 Extended In Java
Using Barcode encoder for Java Control to generate, create USS Code 39 image in Java applications.
QUEUEING NETWORKS
Printing EAN-13 In .NET Framework
Using Barcode generation for ASP.NET Control to generate, create GS1 - 13 image in ASP.NET applications.
Proof The proof is along the same lines as that of Theorem 561 The equilibrium equations of the Markov process {X(t)} are given by
Bar Code Drawer In Visual Studio .NET
Using Barcode encoder for ASP.NET Control to generate, create bar code image in ASP.NET applications.
p(n)
Create Data Matrix ECC200 In Visual C#.NET
Using Barcode creator for .NET framework Control to generate, create Data Matrix ECC200 image in .NET applications.
j :nj >0
Generating Code 128A In VB.NET
Using Barcode creator for Visual Studio .NET Control to generate, create Code 128A image in .NET framework applications.
j =
UPC A Generator In VS .NET
Using Barcode generator for ASP.NET Control to generate, create UPCA image in ASP.NET applications.
j :nj >0 i=1
Encode USS Code 128 In C#.NET
Using Barcode drawer for .NET Control to generate, create Code 128 Code Set C image in VS .NET applications.
p(n + ei ej ) i pij
It suf ces to verify that (5614) satis es the node local balance equations
p(n) j =
p(n + ei ej ) i pij
(5615)
for each j To do so, note that the solution (5614) has the property p(n + ei ej ) = i i j j
p(n)
(5616)
Hence, after substitution of (5616) in (5615), it suf ces to verify that
j =
i i
j j
i pij ,
j = 1, , K
This relation is indeed true since it coincides with the equilibrium equation (5611) This completes the proof A computational dif culty in applying the product-form solution (5614) is the determination of the normalization constant C Theoretically this constant can be found by summing p(n1 , , nK ) over all possible states (n1 , , nK ) However, the number of possible states (n1 , , nK ) such that K ni = M equals i=1 M+K 1 This is an enormous number even for modest values of K and M Hence M a direct summation to compute the constant C is only feasible for relatively small values of K and M There are several approaches to handle the dimensionality problem, including the Gibbs sampler from Section 343 We discuss here only the mean-value algorithm Mean-value analysis The mean-value algorithm is a numerically stable method for the calculation of the average number of customers at station j , the average amount of time a customer spends at station j on each visit and the average throughput at station j The so-called arrival theorem underlies the mean-value algorithm To formulate this theorem, it is convenient to express explicitly the dependency of the state probability p(n1 , , nK ) on the number of customers in the network We write p(n1 , , nK ) = pm (n1 , , nK ) for the network with a xed number of m customers For any state (n1 , , nK ) with n1 + + nK = M the equilibrium probability pM (n1 , , nK ) can be interpreted as the long-run fraction of time
MARKOV CHAINS AND QUEUES
that simultaneously n1 customers are present at station 1, n2 customers at station 2, , nK customers at station K De ne the customer-average probability j (n1 , , nK ) = the long-run fraction of arrivals at station j that see n other customers present at station for = 1, , K Note that in this de nition n1 + + nK = M 1 Theorem 563 (arrival theorem) For any (n1 , , nK ) with j (n1 , , nK ) = pM 1 (n1 , , nK ) Proof By part (b) of Corollary 432,
K =1 n
= M 1,
the long-run average number of arrivals per time unit at station j that nd n other customers present at station for = 1, , K
i pij pM (n1 , , ni + 1, , nK )