Analysis of the asymmetrical shortest two-server queueing model

This study presents the analytic solution for an asymmetrical two-server queueing model for arriving customers joining the shorter queue for the case of Poisson arrivals and negative exponentially distributed service times. The bivariate generating function of the stationary joint distribution of the queue lengths is explicitly determined.


Introduction
The "two-server shortest queueing" model, also known as the "two queues in parallel" model, has obtained quite some attention in Queueing Theory literature, the greater part of it concerning the symrnetrical model. For a short overview of the studies on the symmetrical model, see (12]. The asymmetrical model presents an analytic problem which appeared inaccessible for quite a long time. In the present study, the solution of this problem will be given. The model concerns a two-server system with a Poisson arrival stream of customers with rate >.. An arriving customer joins the shorter queue if the queues are unequal; if they are equal, he joins the queue in front of server i with probability 11" i• i = 1, 2. The server provides customers exponentially distributed service times with service rate l/{3i, i = 1,2. The symmetrical model refers to the case when 11" 1 = 11" 2 = 1/2, P 1 = (3 2 • The analysis of the queueing model requires the investigation of the stochastic process {x 1 (t),:x:z(t),t?: O} with :x,(t) being the number of customers present with server i at time t, i = 1, 2. The analysis of this stochastic process can be reduced to that of a. random walk with state space being the set of lattice points in lli! 2 with integer-valued, nonnegative coordinates.
In applications of the two-server, shortest queueing model, the stationary joint distribution of the queue lengths or its bivariate generating function usually contains all the information required to calculate the various performance characteristics of the model. Approximate techniques have been employed to obtain this information. Usually, they yield the replacement of the infinite state space of the random walk by a finite one. For the resulting process, the Kolmogorov equations for the sta.tiona.ry probabilities a.re then solved numerically. Blanc [4,5), applies the power series algorithm. Here it is assumed that the stationary probabilities can be expressed as a power series made of powers of some suitable chosen function of the traffic load to be handled by the servers. Substitution of these series into the Kolmogorov equations then leads to a set of equations, from which the coefficients of these series can be recursively calculated. The results obtained by this approach are quite satisfactory when compared to those obtained by simulation. Actually, this approach is also based on a special truncation of the state space. Unfortunately, a. sufficient mathematical justification of this approach is still not available. Adan et a.I. [2), and Adan [3] present a.n iterative approach.
Random walks on the lattice in IR 2 with integer valued, nonnegative coordinates are instrumental in the analytical investigation of a large class of queueing models to date. For such random walks and their application to Queueing Theory, quite some information is presently available concerning a fairly general approach of their analysis, cf. [7,8]. For the subclass of semi-homogeneous, nearest neighbor random walks, a more effective analytical approach came recently available, cf. [10,14], and in particular for nearest-neighbor random walks with no one-step transition probabilities to the north, the north-east and the east, cf. [9,[11][12][13]. For this latter case it appeared that the bivariate generating function of the stationa.ry joint distribution of the random walk, if it exists, ca.n be explicitly expressed in terms of meromorphic functions. A meromotphic function is a function which is regular in the whole complex pla.ne, except for at most a finite number of poles in any finite domain.
The random walk to be used in the analysis of the asymmetrical shortest queue is indeed a nearest-neighbor random walk. However, it is not semi-homogeneous, because the one-step transition probabilities at points above the diagonal of the first quadrant differ from those below this diagonal. In fact, this random walk consists of two semi-homogeneous random walks, one at the points above, the other at the points below the diagonal, and they are coupled a.t the points of the diagonal. These two random walks do not have one-step transition probabilities to the north, the north-east and the east and so, it was conjectured that the bivariate generating function of its stationary distribution can be indeed described in terms of meromorphic functions. This conjecture was the starting point of our analysis and it appeared to be true.
We continue this introduction with an overview of the sections of the present study.
In Section 3, properties of the zeros of h 1 (p, r) ::::: 0 and of h 2 ( r, t) = 0 are described. These zeros are denoted by , cf. (3.9), p ± ( r) or r :I: (p) for h 1 (p, r) = 0, r±(t) or t:l:(r) forh 2 (r,t):::::O. (1. 3) The curve h 1 (p, r) = 0, when traced for real p and r, is a hyperbola, which has one of its branches into the first quadrant. A graph with successive vertical and horizontal edges is inscribed into the branch in the first quadrant; see Figure 3.2. The corner points of such a graph correspond to zeros of h 1 (p, r). Analogously, such a graph is inscribed in the branch of h 2 ( r, t) = 0. Any point of h 1 (p, r) = 0 induces also such a graph on h 2 (r, t) = 0 and vice versa. In Figure 3.4, a binary tree is traced, starting at r£ 0 l. Out from rbo) a graph, as mentioned above, is constructed on h 1 (p, r) = 0 and on h 2 (r, t) = 0. Each corner point of such a graph induces on the other hyperbola again such a graph; only ascending graphs are used here. The set of corner points so obtained constitutes the tree generated by r£ 0 J; see for details Section 3. Such a tree, with a properly chosen top rb 0 l, is instrumental in the construction of the functions B( · ), <11 0 ( • ), n,(. ), i = 1, 2.
Section 4 starts with the formulation of a lemma. It states that the functions B(s), c;)) 0 (s) and O;(s), i = 1,2, can all he continued meromorphically into Is I > 1.
For those meromorphic continuations a set of relations is derived, cf. (4.1-4.4). We mention here two of those relations, viz.
It is assumed, cf. assumption 4.2, that all poles of B( s ), <l! 0 ( s ), Oi( s ), i = 1, 2, in j s I > 1 are simple; at a pole of B( · ) its residue is indicated by b( · ); ef> 0 ( • ), wi( · ), stand for residues of 41 0 ( · ), Oi( · ), i = 1,2, respectively. Note that b(s) = 0 implies tha.t s is not a pole of B( · ). From the set of relations ( 4.1-4.4), a set of relations for the residues is obtained; e.g., if r is a pole of B( ·) and also of 4> 0 ( · ), then it is seen from (1.4) that in general p-(r) is a pole of 0 2 ( ·) and r -(r) is that of 0 1 ( • ). The essential point of the analysis is to construct from the set of equations for the residuei; a nonnull solution such that B( · ), i!l 0 , Oi( · ), i = 1, 2, are true meromotphic functions, i.e. the pole set of each of these functions has not a finite accumulation point.
It turns out tha.t there is a nontrivial solution to this problem. The solution of this problem is discussed in Sections 5, 6 and 7.
In Section 5, it is shown that there exists a unique T in T > 1 such that the equa- for the residues b(T) and q, 0 (T) of B( ·) and <I> 0 ( ·) possess a nonnull solution. Actual ly, T is then determined by that zero in T > 1, for which the main determinant oJ the set of equations (1.5) is zero. Note that (1.4) and (1.5) imply that p -(T) is not a pole of !1 2 ( ·) and r -(T) not a pole of !1 1 ( · ).
In Section 6, it is shown that the set of nodes of the tree generated by r& 0 l = T ii the pole set of B( ·) a.s well as of itl 0 ( • ). The poles of U;( · ), i = 1, 2, can be deduced from those of B( ·) and ~0( • ). Further, a recursive set of equations is derived for thi residues at these poles; see Lemmas 6.1 and 6.2. In order to decide whether the po}( set of B( ·) can be used to define a meromorphic function information is required con· cerning the asymptotic: behavior of the poles of B( ·) and their residues b( · ). Thii asymptotic behavior is studied in Section 7. It is shown that this asymptotic beha.v· ior is such that for the pole set of B( · ), a class of meromorphic functions can be con· structed, similarly for 4> 0 ( • ), Oi( · ), i = 1, 2. The elements of these classes of mero· morphic functions are Earametrized BY nonnegative integers mb, mq,1 m; 1 i = 1, 2, bounded from below by Ml, where M is defined by the asymptotic behavior of thi residues, cf. (8.6).
In Section 8, the meromorphic functions B (. ), ~o(. ), n;(. ), i = 1, 2, are intro· duced by using the pole sets obtained in Section 6 for the functions B( · ), <I> 0 ( · ) D;(. ), i = 1, ~ the~ contain the parameters mb, mq,1 mi, i = l, 2, cf. (8.2). Furth_!'.r polynomials B(s), ~0 (s), i = 1,2 are introduced; their degrees are indicated by Nb N 4 ,, N;, i = 1, 2. With the functions so introduced, the functions B( ·) = B( ·) + B ( · ), ~0( ·) = ~(·)+a? ( · ), n;(-) = fi(.) + s\(-), i = 1,2, (1.6 are considered, cf. (8.7). These functions are substituted into equations (1.4), cf (8.9). The asymptotic behavior at infinity of the relations~ obtained is investigat ed. It leads to relations between m and the degrees N of the polynomials Given M, appropriate choices of m d~termines the degrees 'R . With the degree so determined, the functions (1.6) ·~~c again substituted in the. ~quations (1.4). B: considering the resulting equations for properly chosen zero-tu pies of h 1 (p,1") and o ~(r,t), a sysiem o~linear\...nonhomogeneous equations in unknown coefficients of the polynomials B( · ), ' 1> 0 ( • ), O;( • ), i = 1, 2 is obtained. It is shown that this system has a unique solution, and so, the functions of (1.6) are all known and satisfy the equations; moreover, they are all regular in the closed unit disk centered at zero. Theorem 8.1 states that for M ~ 1, a 1 :f. a 2 , the functions so constructed, whenever taking into account the norming condition, determine the bivariate generating functions of the stationary distribution uniquely if and only if al + J > I. The case M = 1 is an 1 2 important practical case. Then the construction of the solution for the case M = 2 is discussed, whereas that for M ;::: 3 is exposed. In Remark 8.2, the meaning of T for the asymptotic behavior of the constructed solution is discussed.
In Section 9, the case a 1 = a 2 , 11' 1 :/:11' 2 , is discussed. For this case, the construction of the· solution of (1.4) is essentially simpler, because here the pole sets do not have a tree structure. In Section 10, relations are derived for some characteristic probabilities and moments of the model. The approach developed in the present study is applicable to a larger class of asymmetrical two-server models, e.g., the asymmetrical variant of the model in [9). This class is characterized by zero one-step transition probabilities to the north, north-east and east in the upper as well as in the lower triangle of the first quadrant [15}.

The Functional Equation
The functional equations for the bivariate generating function of the joint distribution of the queue lengths x 1 , x 2 , of the asymmetrical shortest queue model have been derived in [1]; see Section III.1.2, p. 245. Below we recall these functional equations using mainly the same notation as in [1].
(2.8) By using the properties of the zeros of K 1 (r 1 , r 2 ) when simplifying the expression in square brackets in (2.7) and taking p = T21 r = r1r2 it is readily verified that (2.7) is equivalent to (i) !1 2 (p) + :~,,.-rB(r) +k 1 (p,r)~0(r) = 0, for a zero-tuple (p,r) of h 1 (p,r); \ p I :S 1, I r I :S l; (ii) for a zero-tuple (r, t) of h 2 (r, t); I r I :S 1, It I S l; note that the derivation of (2.9) (ii) is analogous to that of (2.9)(i). From the above, it is seen that the determination of the bivariate generating function E{r: 1 r; 2 } of the stationary joint distribution of the queue lengths requires the construction of functions S1 1 (p ), n 2 (p ), ~0 (p) and B(p ), which should satisfy the following conditions: (i) they are regular for Ip I < 1 and the sum of the coefficients in their series expansions in powers of pn converges absolutely; (2.10) (ii) they satisfy relations (2.1) (iv), (2.5), (2.9) (i),(ii) and, cf. (2.1), ni(O) == ~0 (0), i = 1, 2.
[3], p. 25, it follows that the queue length process (x 1 (t),~(t)), of which the state space is irreducible, is positive recurrent and further that there is only one solution, which satisfies (2.10) and the norming condition. Hence it suffices to construct functions B( · ), 0 1 (.) and nl (. ), which satisfy (2.10).

On the Zeros of h 1 (p,r) and h 2 (r,t)
In this section we shall describe several properties of the zeros of h 1 (p, r) and h 2 (r, t) and introduce several functions of these zeros; these functions are needed to describe the functions 0 1 ( r ), D 2 (p ), <1> 0 ( r) and B( r ). Because of the symmetry between h 1 (p,r) and h 2 (r,t), we mainly restrict the discussion to h 1 (p,r); those for h 2 (r,t) follow by interchanging a 1 and a 2 .
Analogously, we define the ladders l(tn) and r (tn)· Remark 3.2: Note that every point of h 1 (p, T) with p > 1 or T > 1 induces a ladder on h 2 (r, t) and visa versa.
(3.13) Next we introduce a notation to describe all the ladder points on the up-ladders on h 1 (p,t) = 0 as well as on h 2 (r,t) = 0 generated by a point Define form= 0, 1, 2, ... , the binary numbers The tree generated by (r 0 ,p 0 ) is defined as follows: its nodes at the nth level, n = 0, 1,. .. , arc In Figure 3.4, we depicted the nodes at levels O, 1, 2 and 3 for the tree generated by From the definitions above it is readily seen that the tree so constructed contains all the ladder points on h 1 (p,r) = 0 and h 2 (r, t) = 0 generated by To with f: 1 (p 0 ,r 0 ) = O; note that l(r 0 ) is the set of nodes on the left branch of the tree and I ( r 0 ) is the set of nodes on the right branch of the tree. The tree generated by (t 0 ,r 0 ) is defined analogously, interchange the symbols r and t and also p and r.
Note that, cf. Lemma 3.1, for j = 1, 2, In this section we shall consider the meromorphic continuation of the functions 11 1 (s), Ois), B(s) and <1> 0 (s) out from Is I ~ 1 into Is I > 1.
Lemma 4.1: The functions n 1 (s), 0 2 (s), B(s) and <I> 0 (s) can be conlinued meromorphically out from J s I S 1 into I s I > 1.
For the proof of this lemma, see Appendix B.
Remark 4.1: The lemma does not imply that these continuations are meromorphic functions, i.e., have only a. finite number of poles in every finite domain, but it implies that their only singularities are poles or accumulation points of poles.
Remark 4.4: Concerning the introduction of the latter assumption, it is noted that Remark 4.2 also applies here.

The Equation for the Top of the Tree
In this section, we derive a relation for the smallest in absolute value pole of B( ·) and of ~0 ( • ).

From (4.3) and (4.4) we have that for
From (2.3) and (3.7), it is seen that aba -7l'1r-(1)-7r2P-(l)>O. (ii) has in r > 1 a unique root r = T, of multiplicity one, for J + J > 1, it is given by Proof: It is readily seen that the above equation is equivalent to and because the right-hand side of (5. 7) is readily seen to be larger than 1 for r ~ 1 and it is continuous and increasing in r with a finite limit for r-oo, the first statement of the lemma follows. It is simply verified that r = T, p -( r) = p -(T), r -( r) = r -(T) satisfy (5. 7). It remains to show that, cf. (3.1) (ii) and Lemma 3.1, readily seen that (5.8) (ii) holds. D Assumption 5.1: Henceforth, it will be assumed that, cf. also Assumption 6.1.
Remark 5.1: Concerning the introduction of the latter assumption it is noted that Remark 4.2 also applies here.
Since r =Tisa simple zero of the determinant of (5.3), it follows from (5.2), and Assumption 5.1 that r =Tisa simple pole of B(r) and also of IP 0 (r). (5.9) In Section 8, Remark 8.2, it will be shown that T is the smallest pole of R( ·) and also of ~0 ( · ). From (4.G), (5.2) and Assumption 5.1, we obtain, note that these relations are linearly dependent.
Relations (6.4) and (6.8) represent the relations for the poles and residues at the zerolevel of the tree, cf. also (6.3); the relations (6.10), (6.11), (6.13) and (6.14) describe the relations for the poles and residues at the first level of the tree generated by r& 0 l.
To obtain those relations at the nth level of the tree, we introduce, cf. (6.9), the following.
We consider first the case of 6 being even.
It remains to considet the hypothesis, cf. (6.18), that w 2 (p~")) for o even and w 1 (r~")) for 6 odd are both finite and nonzero. By induction, it is seen from (6.20) and from (6.22) for n = 1,2, ... , and (6.10), (6.11), (6.13), and (6.14) that these hypotheses are indeed valid. Note that these relations show that all residues are zero or no one is zero; the first case is impossible, see the penultimate paragraph of Appendix n.
Proof: The proof follows immediately from the above analysis in this section. D Lemma. 6.1 describes the equations for the residues at all nodes of the tree generated by Tbo) == T. But, as we have seen in Section 3, every node T~n), with 6 even, induces on h 2 (r,t) = 0 a ladder, and analogously for 6 odd, a ladder on h 1 (p,r) = 0, see The set of relations (6.27), (6.28}. and (6.29) is insufficient to determine the unknown residues wi(r;), b(t;) and efi 0 (f;)· However, Assumption 4.1 leads to the conclusion that the only solution of this set of relations is the zero-solution, i.e., at all induced down-ladders, the residues at the elements of these down-ladders are zero, so that these elements cannot be poles of B( · ), 4> 0 ( • ), D 1 ( ·) and 0 2 ( · ).
To see this, first note that every element of the tree generated by r& 0 ) = T induces down-ladders on h 1 (p, r) = 0 or on h 2 (r, t) == 0, and elements of down-ladders again induce down-ladders. Since the tree generated by r& 0 ) consists of an infinite but countable number of nodes, it follows that the finite part of h 1 (p, r) = 0 with p E [p -(1 ), p + (1 ) For further analysis of the functions B( · ), ~0( · ), i1 1 ( ·) and r! 2 ( · ), we require the asymptotic behavior for n~oo of the residues of these functions at their polesi see the preceding section.

t
The lemma. a.hove describes only the asymptotic beha.vior of the va.rious residues a.1 those nodes of the tree generated by (T,p-(T)), which belong to the left-or rightmost branch of that tree, i.e., at the nodes To consider the asymptotics for n-+oo of the residues at a generic point T~m+n), 6E{0 1 l 1 ... 1 2m+n_1}, we write (7.9 Bence, r~m+n), n=0,1,2 1 ••• , is the tree generated by r~m), and it is a. subtree o tha.t generated by T. Now, write dn as a binary number a.n'lt in this binary represen ta.tion denote by d~n) the number of zeros, (7.10 d~n) the number of ones, BO that (7.11 It readily follows from (7.6), (7.9) and (7.10) that for every finite n: for m-+oo, be a generic element c (7.13 (iii) lim m-+oo with em,dn,d~n),d~n), for given 8E{O,l, .. .,2m+n_l}, as defined in (7.9) and (7.10), and, cf. (6.17), :::: r-(r~m + n)) for 8 odd.
Proof: For 6:::: em2n + d consider, the tree generated by rim), which is a subtree of the tree generated by r& 0 ) = T. Apply for this subtree LeTuma 7.1 with n = 1.
Note that next to these conditions we have the two conditions which stem from the definitions (2.1); see also (2.10). Further, it should be mentioned that the set of Kolmogorov equations, which are equivalent to the conditions (2.10), contains one dependent equation. So, in total, the coefficients of the polynomials have to satisfy 'ii 1 +1 + /1-i + 1 + 2-1 = }l 1 + 'j]. 2 + 3 conditions. Consequently, we have, cf. (iv) 'ji 1 +'jl 2 +3 conditions ha.veto be satisfied.
Obviously, we have quite some freedom in choosing the exponents in (8.2) and the degree of the polynomials in (8.7). This freedom is not so surprising because in general, a meromorphic function does not have a unique decomposition, ( cf. [15], p. 304); see also Remark 8.1.
The available freedo~ will be used to choose the numbers in (8.14) (i) and (ii) as small as possible, with M being defined in (8.6). Before discussing this point, we first consider several zero-tuples which are most appropriate for the determination of the polynomials in (8. 7).
(8.29) From (3.4), it is seen that p( r +) is a zero with multiplicity two of h 1 (p, r + ). Consequently, it follow from (4.3) that p(r+) should be a zero of multiplicity two of (4.3) with r = r +, since T = T + is not a pole of B(r) and <Ii 0 (r), and p = p(r +)is not a pole of !1 2 (p). Hence, from (2.9): Note that p(r + )r + =!= 0.
We proceed with the determination of the polynomials in (8. 7). With regard to the available freedom mentioned above we shall try to choose the degrees of the polynomials in (8.7) as small as possible.
First we consider the case ' M =i.  (iv) n 1 (r) = -f: :L .p~n) + f: E "' ls) _ r I r I < r+ (T). n=O aE':Ilnr~n) n=l JE<:.Bnr-r~n) ,.~n)' These functions have meromorphic continuations throughout the whole comv.lex plane, which are given by the right-hand sides of (8.39) (i) ... (iv). The residues b~n), q)~n), w~s), i = 1, 2 can be calculated recursively (see Lemma 6.2), they all contain the factor q) 0 ( rb 0 )), which is uniquely determined by a~O.i(l) + J 2 n1(1) = J 1 +ck-1. , that the radius of convergence of the latter generating function is larger than one, analogously, for E{r21(~ = x 1 +1)}. From (2.4) and (8.39), the bivariate generating functions E{r~lr~(~ > x 1 }} and E{r;lr;2(x 1 > x 2 )} are obtained. It is readily seen that the domain of convergence of these bivariate generating functions contains the Cartesian product unit disks I r 1 I 5 1, I r 2 I ~ 1, as a true subset. Hence, the coefficients in the series expansions of the bivariate generating function E{r~lr~2} is an absolutely convergent solution of the Kolmogorov equations. Hence, Foster's criterion, cf. Remark 2.2, implies that the queue length process {x 1 (t),.xi(t), t > 0} is positive recurrent for J + 1 > 1; (2.6) shows that this condition is also necessary. Because all generating ruJctiofis contain efi 0 (T) as a linear factor, cf. Lemma 6.1, this factor follows from (8.40), cf. (2.1) and (2.5), and so it is uniquely defined. For the uniqueness of the solution constructed for the conditions of the theorem, see Remark 2.2. 0 Next, we consider the case This choice is again consistent with (8.14) and it follows that fit= 'ii2=1. Remark 8.1: The degrees of the polynomials and the exponents of the meromorphic functions have been introduced in (8. 7) and (8.2). They have to be determined in such a way that (8.14) is satisfied and Fl(T)+Fi-(r), i = 1,2 ate zero at P:;+l points. In this determination there is no objection replacing mb by mb + hb, m<P by Tf!.c/>+ht/>, m 2 by m 2 +h 2 and m 1 by m 1 +h 1 , with hb,hq,,h 2 ,h 1 , positive integers (and M defined by (8.6) ). Such a change when compared with the case that hb, hq,, h 2 , h 1 are all zero, actually amounts to subtraction of a polynomial from the meromorphic function and addition of that polynomial to the "/\" polynomial; see (8.7). In fact, this also occurs by noting thaL the solution given by (8.44) also holds for the case .M =i. i Remark 8.2: From (8.2) and (8.7), it is readily seen that T = r6°) = (<: 1 + a 1 2 ) > 1 is the smallest pole of tl? 0 ( · ) and also of B( · ). Hence, T determines the asymptotic behavior of Pr{x 1 = x 2 = n} for n-.oo, i.e.
Similarly, it is seen that are the smallest poles of G 2 ( ·) and 0 1 ( • ), respectively, and so they determine the leading term in the asymptotic behavior of Pr{:xi = n,x 1 = O} and Pr{x 1 = n,X:J:::: O} for n-oo. since for M = 0, the first sum in the right-hand side converges absolutely and thesecond sum is a well-defined meromorphic function, analogously for the other sums in (8.39).
If a 2: 2 no stationary joint distribution exists.
10. Some Expressions for Probabilities a.nd Moments, a 1 ;j;; a 2 In this section, we derive some expressions for several characteristics of the queue lengths.
We consider first the case (10.1) since we have to discuss separately the case that one of the a;'s is equal to one. From(2.4) and Appendix D, we have The summation of (10.5), (ii)-( iv) yields the expression for E{xi}. The expression for E{x 2 } then follows by interchanging a 1 and a 2 and changing the signs of the terms containing B( · ).
(10.6) l 2 By noting that relations (10.3) (ii)-(iv) and (10.4) have been all derived from (d.6), in which 1 -a 1 does not occur, it is seen that these relations also apply to the present case with a 1 = 1.
The singularities of <P 0 ( r) in I r I > 1 can only be poles, because k 1 (p, r ± (p )) and k 2 (p±(r),r) arc regular in jpj >1 and jrJ >l, respectively; note (3.4), and similarly for the other coefficients in (b.3)-(b.6). Further <I? 0 (r) has at least one pole in { T: 1 < I r ! :::; oo }, because if <t> 0 ( r) would be regular here, then, since it is also regular for I r I < 1, cf. (2.10), it is necessarily a constant, as Liouville's theorem implies. Analogously, for B(r), D 2 (p) and 0 1 (r). Consequently, Lemma 4.1 is proved. 0 From the analytic continuations discussed above, it is seen that the relations (b.3)-(b.6) hold for all those r, t,p and r where the functions in (b.3)-(b.6) are finite. Consequently, it is seen that the validity of the relations (4.1)-(4.4) has been established.

Appendix C
The integer M ?.. 0 has been defined ,Ln (8.6). Numerical results indicate that M is always larger than zero. A proof of M > 0 seems to be rather lengthy and intricate.
Below we discuss some cases, for which the proof is fairly simple.
Finally, consider the case 7r 1 = 71" 2 =!and a 2 !0. It is readily verified that R{ -+0 for a 2 LO and so it is seen from ( c.5) that for a. 2 sufficient small, the numerator Jn the last term of ( c.5) will be negative and it follows again that >. 1 µ 1 < -1 and so M > 0, since >. 2 µ 2 < O.
However, for n sufficiently large, this inecpality cannot hold for any b; > 1, i = 1,2.