## 1. Introduction

In [6] N. Bailey considered a queueing model with a bulk service admitting batches with a maximum size . This model describes several situations where the customers are admitted in groups, for example transportation of people in vehicles with a maximum passenger capacity. In our case we assume that batches have a fixed size (see [24, Chapter 4]

). We have an example of these situations in some shuttle routes where their buses only start their trip if they complete the vehicle’s capacity. We can find more examples of this behavior in certain bureaucratic process, for instance this happens in some public museums where the guides only admit a fixed number of customers and if this number is not completed they don’t work. Let us assume that an individual arrives at each epoch of Poisson occurrence with rate

, and the service only starts if the queue size reaches or exceeds . The service time distribution of a batch is assumed to be exponential with parameter . In terms of Kendall’s notation [20] used in [24], we say that our system is .This system corresponds to a stationary Markov process with path function whose transition probability function

satisfies the following conditions

(1.1) |

When we are in the classical case of a Birth-and-Death process (see [19]). Birth and Death processes have been used to model any M/M/1 system (see [5, Chapter III]). As in [10, Chapter XVII]), considering (1.1) with the Chapman-Kolmogorov equations, we obtain the following system of infinite differential equation with :

Let be a infinite matrix function , then is the solution of the following initial matrix value problem

(1.2) |

where is such that for each :

schematically

denotes the transpose of matrix .

According to W. Sternberg (see [29]) the problem (1.2) has a unique solution. In [19], S. Karlin and J. McGregor arrived at an integral expression for the entries of in the case of , which is a Birth-and-Death process. They used some aspects of the orthogonal polynomial theory. Using Karlin and McGregor’s approach, T. S. Chihara and M. E. H. Ismail (see [9]) connected sequences of orthogonal polynomials with a queueing model. Later on in [17] M. E. H. Ismail revisited this connection. In this paper we also use Karlin and McGregor’s method to study our case for general , using now properties of multi-orthogonal polynomials.

Let be a sequence of polynomials generated by the recurrence relation

(1.3) |

with initial conditions

which implies that

We consider a family of vector polynomials

that satisfy the recurrence relation(1.4) |

Given two complex numbers and we denote by the section of straight line that connects with . containing the endpoints. The following Theorem is proved in Section 4:

###### Theorem 1.1.

There exists a system of weights , defined on the starlike set

such that:

(1.5) |

where

denotes Kroneker’s delta.

In the proof of above Theorem 1.1 we follow ideas published in the papers [3, 4]. Let us state our main result:

###### Theorem 1.2.

The matrix function whose entries are

(1.6) |

is the unique solution of the initial value problem (1.2).

In Section 3 we obtain expressions for the polynomials , and , and in Section 4 we give more details about the structure of the weights

. Such expressions are not easily manejable. We think that an extension of a Golub-Welsch algorithm would be very interesting to estimate the integral that appears in (

1.6) using the recurrence relations in (1.3) or (2.3), and so avoid the computation of the weights . Despite this work does not give a computation algorithm, in Section 6, we give an appropriate convergent scheme of quadrature rules.## 2. Bi-orthogonality

In this Section we review some concepts and simplify results from [3, 4] to accommodate their statements and notation to our special case. They are necessary to prove Theorem 1.1 and Theorem 1.2 in Section 4. We consider the family of polynomials

(2.1) |

Then the elements of the family satisfy the following recurrence relation

(2.2) |

with initial conditions

We see

This kind of families of polynomials has been studied in several publications such as [3, 4, 21, 22, 23]. We also introduce the vectors of polynomials that satisfy the recurrence relation

(2.3) |

We have that

(2.4) |

The recurrence relations (2.2) and (2.3) are both associated to the infinite non-symmetric matrix

which determines an operator in the space of sequences . Set the standard basis of . is defined as follows

Let be a polynomial with an arbitrary degree . Then , denotes the operator:

###### Lemma 2.1.

Consider the sequence of polynomials satisfying the recurrence relation (2.2). Then

###### Proof.

We recall that for each we have that , then . This means that

We now consider . For each , let us assume that . By the recurrence relation (2.2), we have that

hence

The proof is completed by induction. ∎

Given an arbitrary vector polynomial we denote for every system with , the ”dot” product

###### Lemma 2.2.

Consider the sequence of vector polynomials satisfying the recurrence relation (2.3) and denote . Then

###### Proof.

We observe that for each

Let us consider . For each , let us assume that . By the recurrence relation (2.3), we have that

This implies that

The proof is completed by induction. ∎

In we consider the regular inner product. This is that given two and elements of

where represents the complex conjugate transpose operator of . When is real .

## 3. An algebraic equation: some properties

In [4] the following the algebraic equation is sudied:

(3.1) |

We enumerate some needed properties of its multivalued solution , which is an algebraic function of order and genus . Its inverse rational function

(3.2) |

gives the composition of a conformal map of the Riemann sphere to a Riemann surface of the function and the projection of to the complex plane. In order to describe this surface we introduce certain concepts and notations, and also review some results stated in [4]. Given a complex numbers we denote by with , the ray starts at with slope . Set

Let us declare the following sets:

(3.3) |

We also introduce

According to [4, Proposition 1] there exist global branches of in (3.1) such that

and

(3.4) |

In the inequalities are strict.

Let us describe then the Riemann surface associated to the multi-valued function such that the sheet structure corresponds to the choice of branches as above in (3.4). Let denote the projection of on , with (multivalued) representing its inverse branches. We have that

We also denote a closed contour in which separates and . The contour is orientated such that lies on the left, this is positive side. We take

Hence

We now sketch the proof of [4, Proposition 1] where some properties of the multivalued function in (3.1), are stated and they are needed for our purposes. From Vieta’s relations in (3.1) we have that the solutions of satisfy that

(3.5) |

Combining (3.2) and (3.5) we have that (multivalued function) has a branch point at infinity of order . Then we suppose that ’s branches, denoted by , have the following behavior

(3.6) |

First we analyze for . Notice when . The inverse function (see (3.2)) is decreasing in the interval and increasing on . This means that there are two positive branches of in . We denote them by and such that

When is even there also is a negative branch of on , that we denote by . Then we can introduce the notation

(3.7) |

Let us define an analytic continuation of from to . Consider a solution of (3.1), then the following equality holds

So for any fixed there exists a choice of branch , satisfying the symmetric relation:

We choose ,

(3.8) |

Using Monodromy Theorem (see [1, Chapter I 15]), we extend analytically the branch from a neighborhood of to the domain . We declare . Since (3.8) is satisfied near for , then

Taking into account the behavior at infinity in (3.6), the branch has an analytic continuation from to the simple connected domain

Starting from above set and using (3.8) with we extend analytically as follows

Since , we have that

So combining the above identity with (3.8) we have that satisfies the jump condition

where

and is the closure of . Hence .

We proceed analogously with the remaining branches of . Consider the fundamental sector of

We extend the branches to using the symmetry relations in (3.8):

(3.9) |

A branch with even index, , is defined in as a direct analytic continuation of the branch to the domain from the upper part of the boundary of , and to the domain from the lower part of the boundary of . While a branch with odd index, , is defined on taking the analytic continuation of the branch to from the lower part of the boundary of , and to from the upper part of the boundary of .

Consider the domain

and define the function as follows

Let us recall (3.7) and (3.8). If the maximum principle to the holomorphic function we can extend the inequality in to and therefore to the whole complex plane . Analogously we can extend all the relations stated in (3.7) to and obtain (3.4).

Once we have sketched the proof of the relations (3.4), we define

(3.10) |

Using (3.4) we observe that the following functions defined on , are real valued and never change sign.

(3.11) |

So given ,

where

Since never vanishes in and taking into account (3.6) we have that

(3.12) |

Now using (3.11) and Sokhotski–Plemelj (see [14, Chapter I, equality (4.8)]) we obtain that:

(3.13) |

Let us express the polynomials satisfying the recurrence relation (2.2) and the vector polynomials in (2.3), in terms of the solutions of the equation (3.1).

###### Theorem 3.1.

Fix . The polynomial , can be expressed

(3.14) |

Given , the vector polynomial has the form

with

(3.15) |

###### Proof.

In (3.14) we follow the steps of [3]. We observe that the following function satisfies the recurrence relation (2.2)

The coefficients are defined from the initial conditions:

From Cramer’s rule we find

(3.16) |

Using now Vieta’s formulas in (3.5), we arrive at

We also have that

Then

Consider the functions

Then the denominator of in (3.16) can be written as:

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