### Proof of Lemma 7

Without loss of generality we can take \(j=1\). Instead of performing the summation in (7) over the whole group \(S_{K},\) we partition \(S_{K}\) into \(\left( K-2\right) !\) subsets \(A_{\pi }\), and perform the summation over each \(A_{\pi }\) separately. If the result will not depend on \(\pi ,\) we are done. Here \(\pi \in S_{K},\) and, needless to say, for \(\pi ,\pi ^{\prime }\) different we have either \(A_{\pi }=A_{\pi ^{\prime }}\) or \(A_{\pi }\cap A_{\pi ^{\prime }}=\varnothing .\)

Let us describe the elements of the partition

\(\left\{ A_{\pi }\right\} .\) So let

\(\pi \) is given, and the string

\(i_{1},i_{2},i_{3},\ldots ,i_{l},i_{l+1} ,\ldots ,i_{K}\) is the result of applying the permutation

\(\pi \) to the string

\(1,2,\ldots ,K.\) Then we include into

\(A_{\pi }\) the permutation

\(\pi ,\) and also

\(K-1\) other permutations, which correspond to the cyclic permutations, e.g. we add to

\(A_{\pi }\) the strings

$$\begin{aligned}&i_{K},i_{1},i_{2},i_{3},\ldots ,i_{l},i_{l+1},\ldots ,i_{K-1},\\&i_{K-1},i_{K},i_{1},i_{2},i_{3},\ldots ,i_{l},i_{l+1},\ldots ,i_{K-2}, \end{aligned}$$

\(i_{K},i_{1},i_{2},i_{3},\ldots ,i_{l},i_{l+1},\ldots ,\) and so on. We call these transformations ‘cyclic moves’. Now with each of

*K* permutations already listed we include into

\(A_{\pi }\) also

\(K-2\) other permutations, where the element

\(i_{1}\) does not move, and the rest of the elements is permuted cyclically, i.e., for example from

$$\begin{aligned} i_{K-1},i_{K},i_{1},i_{2},i_{3} ,\ldots ,i_{l},i_{l+1},\ldots , \end{aligned}$$

we get

$$\begin{aligned} i_{K},i_{2},i_{1},i_{3},\ldots ,i_{l} ,i_{l+1},\ldots ,i_{K-1};\ i_{2},i_{3},i_{1},\ldots ,i_{l},i_{l+1},\ldots ,i_{K-1} ,i_{K}, \end{aligned}$$

and so on. We call these transformations ‘restricted cyclic moves’. The main property of thus defined classes of configurations is the following: Let

\(a\ne b\in \left\{ 1,2,\ldots ,K\right\} \) be two arbitrary indices, and

\(l\in \left\{ 2,\ldots ,K\right\} \) be an arbitrary index, different from 1. Then in every class

\(A_{\pi }\) there exists exactly one permutation

\(\pi ^{\prime },\) for which

\(i_{1}=a\) and

\(i_{l}=b.\)Given \(\pi ,\) take the customer \(l\ne 1,\) and its destination, *D*(*l*). If we already know the position \(i_{1}\) of customer 1 on the circle \(C_{K},\) then in the class \(A_{\pi }\) there are exactly \(K-1\) elements, each of them corresponds to a different position of the server *l* on \(C_{K}.\) If it so happens that \(i_{1}=D\left( l\right) ,\) then for no position of the server *l* the transit from *l* through \(i_{1}\) happens. The same also holds if \(i_{1}=\left( D\left( l\right) +\frac{K-1}{2}\right) {\text {mod}} K\) or \(i_{1}=\left( D\left( l\right) +\frac{K+1}{2}\right) {\text {mod}}K.\) For all other \(K-3\) values of \(i_{1}\) the transit from *l* through \(i_{1}\) happens precisely for one position of *l* (among \(K-1\) possibilities). Totally, within \(A_{\pi }\) we have \(\left( K-1\right) \left( K-3\right) \) transit events. Since \(\left| A_{\pi }\right| =K\left( K-1\right) ,\) the lemma follows.

### Proof of Lemma 9

Again we partition the group \(S_{KL}\) into cosets. Each coset \(A_{\pi }\) has now \(K\left( K-1\right) L\left( L-1\right) \) elements. With every permutation \(\pi \) of the *KL* points of the discrete torus \(\mathcal {T}_{KL}\) we first include in \(A_{\pi }\) all its ‘2D cyclic moves’, i.e. permutation \(\pi \) followed by an arbitrary shift of \(\mathcal {T}_{KL}\); there are *KL* of them. Also, with each permutation \(\tilde{\pi }\) from \(A_{\pi }\) we include in \(A_{\pi }\) all \(\left( K-1\right) \left( L-1\right) \) permutations which are obtained from \(\tilde{\pi }\) by performing a pair of ‘independent restricted cyclic moves’: one of them cyclically permutes all the sites of the meridian of the point \(\pi \left( 0,0\right) ,\) and another cyclically permutes all the sites belonging to the parallel of the point \(\pi \left( 0,0\right) .\) There are \(\left( K-1\right) \left( L-1\right) \) such independent restricted cyclic moves. All other \(\left( K-1\right) \left( L-1\right) \) points of the torus \(\mathcal {T}_{KL},\) as well as the point \(\pi \left( 0,0\right) ,\) stay fixed during these independent restricted cyclic moves. The idea behind this definition is to ensure the following property: suppose we know for the permutation \(\pi \) that a point \(\pi \left( k^{\prime },l^{\prime }\right) \) belongs to the parallel of the point \(\pi \left( 0,0\right) ,\) while the point \(\pi \left( k^{\prime \prime },l^{\prime \prime }\right) \) belongs to the meridian of \(\pi \left( 0,0\right) .\) Then for any three different points \(a,b^{\prime } ,b^{\prime \prime }\in \mathcal {T}_{KL},\) such that \(b^{\prime }\) belongs to the parallel of *a*, and \(b^{\prime \prime }\) belongs to the meridian of *a*, there exists exactly one permutation \(\bar{\pi }\in A_{\pi },\) such that \(\bar{\pi }\left( 0,0\right) =a,\) \(\bar{\pi }\left( k^{\prime },l^{\prime }\right) =b^{\prime }\) and \(\bar{\pi }\left( k^{\prime \prime },l^{\prime \prime }\right) =b^{\prime \prime }.\)

Now we fix one class \(A_{\pi },\) and compute the number of transits to node \(\bar{\pi }\left( 0,0\right) \) for all \(\bar{\pi }\in A_{\pi }.\) Without loss of generality, and in order to simplify the notation, we consider only the case when \(\pi \) is the identity \(e\in S_{KL}.\) We denote the permutations from \(A_{e}\) by the letter \(\varkappa .\)

Clearly, the transits from node \(\varkappa \left( k,l\right) \) to \(\varkappa \left( 0,0\right) \) can happen only if either *k* or *l* are 0. So let us fix some integer \(k\in \left\{ -\frac{K-1}{2},\ldots ,\frac{K-1}{2}\right\} ,\) \(k\ne 0,\) and let us count the number of possible transits from \(\varkappa \left( k,0\right) \) to \(\varkappa \left( 0,0\right) \) while \(\varkappa \) runs over \(A_{e}.\) Without loss of generality we can assume that the destination \(D(k,0) =\left( 0,0\right) .\)

As we said already, as \(\varkappa \) runs over \(A_{e},\) the node \(\varkappa \left( 0,0\right) \) can be anywhere on the torus \(\mathcal {T}_{KL}.\) The node \(\varkappa \left( k,0\right) \) can be then anywhere on the parallel of \(\varkappa \left( 0,0\right) .\) If \(\varkappa \left( 0,0\right) =\left( 0,0\right) \left( =D\left( k,0\right) \right) ,\) then no transit to \(\varkappa \left( 0,0\right) \) can happen, independently of the location of \(\varkappa \left( k,0\right) .\) The same is true when \(\varkappa \left( 0,0\right) =\left( x,y\right) \) with \(x=\pm \frac{K-1}{2},\) \(-\frac{L-1}{2}\le y\le \frac{L-1}{2}.\)

For every of the \(\left( K-3\right) \) locations \(\left( x,0\right) \in \mathcal {T}_{KL}\) of the node \(\varkappa \left( 0,0\right) \)—namely, for \(x=-\frac{K-1}{2}+1,\ldots ,-1,+1,+2,...\frac{K-1}{2}-1\) we have one transit per location (or, more precisely, \(L-1\) transits per location, due to the restricted cyclic moves along the meridian).

For every of the location \(\left( 0,y\right) \) of the node \(\varkappa \left( 0,0\right) \)—namely, for \(y=-\frac{L-1}{2},\ldots ,-1,+1,+2,\ldots ,\frac{L-1}{2}\) we have \(2\times \frac{1}{2}=1\) transits per location (or, again more precisely, \(L-1\) transits per location), since there can be two transit events, each with probability \(\frac{1}{2}.\)

For any other remaining location of the node \(\varkappa \left( 0,0\right) \)—and there are \(\left( K-3\right) \left( L-1\right) \) of them, we get \(\frac{1}{2}\) of transit per location (more precisely, \(\frac{L-1}{2}\) transits per location).

Summarizing, we have totally \(\left[ \left( K-3\right) +\left( L-1\right) +\frac{1}{2}\left( K-3\right) \left( L-1\right) \right] \left( L-1\right) \) transits from the node \(\varkappa \left( k,0\right) \) to the node \(\varkappa \left( 0,0\right) \), as \(\varkappa \) runs over \(A_{e}.\) And there are \(\left( K-1\right) \) such nodes.

All in all, we have

$$\begin{aligned}&\left[ \left( K-3\right) +\left( L-1\right) +\frac{1}{2}\left( K-3\right) \left( L-1\right) \right] \left( K-1\right) \left( L-1\right) \\&\quad +\left[ \left( L-3\right) +\left( K-1\right) +\frac{1}{2}\left( L-3\right) \left( K-1\right) \right] \left( L-1\right) \left( K-1\right) \end{aligned}$$

transits, so the probability in question is given by

$$\begin{aligned}&\frac{\left( K-3\right) +\left( L-1\right) +\frac{1}{2}\left( K-3\right) \left( L-1\right) +\left( L-3\right) +\left( K-1\right) +\frac{1}{2}\left( L-3\right) \left( K-1\right) }{KL}\\&\quad =\frac{3\left( K-3\right) +3\left( L-3\right) +\left( K-3\right) \left( L-3\right) +4}{KL}\\&\quad =\frac{3K+3L+\left( K-3\right) \left( L-3\right) -14}{KL}\\&\quad =\frac{KL-5}{KL}. \end{aligned}$$

### Proof of Theorem 14

Using the results (and notation) of [

3], we get that the NLMP is the evolution of the measure

\(\otimes \mu _{v}\) on the states (queues

\(q_{v}\)) of the servers at the nodes

\(v\in \mathbb {Z}^{1},\) given by the equations

$$\begin{aligned} \frac{d}{dt}\mu _{v}\left( q_{v},t\right) =\mathcal {A}+\mathcal {B} +\mathcal {C}+\mathcal {D}+\mathcal {E} \end{aligned}$$

(24)

with

$$\begin{aligned} \mathcal {A}=-\frac{d}{dr_{i^{*}\left( q_{v}\right) }\left( q_{v}\right) }\mu _{v}\left( q_{v},t\right) \end{aligned}$$

(25)

the derivative along the direction

\(r\left( q_{v}\right) \) (in our case, since we assume exponential service times with rate 1, we have

\(\frac{d}{dr_{i^{*}\left( q_{v}\right) }\left( q_{v}\right) }\mu _{v}\left( q_{v},t\right) =\mu _{v}\left( q_{v},t\right) \)),

$$\begin{aligned} \mathcal {B}=\delta \left( 0,\tau \left( e\left( q_{v}\right) \right) \right) \mu _{v}\left( q_{v}\ominus e\left( q_{v}\right) ,t\right) \left[ \sigma _{tr}\left( q_{v}\ominus e\left( q_{v}\right) ,q_{v}\right) +\sigma _{e}\left( q_{v}\ominus e\left( q_{v}\right) ,q_{v}\right) \right] \end{aligned}$$

(26)

where

\(q_{v}\) is created from

\(q_{v}\ominus e\left( q_{v}\right) \) by the arrival of

\(e\left( q_{v}\right) \) from

\(v^{\prime }\), and

\(\delta \left( 0,\tau \left( e\left( q_{v}\right) \right) \right) \) takes into account the fact that if the last customer

\(e\left( q_{v}\right) \) has already received some amount of service, then he cannot arrive from the outside;

$$\begin{aligned} \mathcal {C}=-\mu _{v}\left( q_{v},t\right) \sum _{q_{v}^{\prime }}\left[ \sigma _{tr}\left( q_{v},q_{v}^{\prime }\right) +\sigma _{e}\left( q_{v} ,q_{v}^{\prime }\right) \right] , \end{aligned}$$

(27)

which corresponds to changes in queue

\(q_{v}\) due to customers arriving from other servers and from the outside (in the notation of (

1),

\(\sigma _{e}\left( q_{v},q^{v}\oplus w\right) =\lambda _{v,w}\));

$$\begin{aligned} \mathcal {D}=\int _{q_{v}^{\prime }:q_{v}^{\prime }\ominus C\left( q_{v}^{\prime }\right) =q_{v}}d\mu _{v}\left( q_{v}^{\prime },t\right) \sigma _{f}\left( q_{v}^{\prime },q_{v}^{\prime }\ominus C\left( q_{v}^{\prime }\right) \right) -\mu _{v}\left( q_{v},t\right) \sigma _{f}\left( q_{v},q_{v}\ominus C\left( q_{v}\right) \right) , \end{aligned}$$

(28)

where the first term describes the situation where the queue

\(q_{v}\) arises after a customer was served in a queue

\(q_{v}^{\prime }\) (longer by one customer), and

\(q_{v}^{\prime }\ominus C\left( q_{v}^{\prime }\right) =q_{v},\) while the second term describes the completion of service of a customer in

\(q_{v}\);

$$\begin{aligned} \mathcal {E}=\sum _{v^{\prime }\text {n.n.}v}\beta _{vv^{\prime }}\left[ \mu _{v^{\prime }}\left( q_{v},t\right) -\mu _{v}\left( q_{v},t\right) \right] , \end{aligned}$$

(29)

where the

\(\beta \)-s are the rates of exchange of the servers.

For the convenience of the reader we repeat the Eqs. (

24)–(

29) once more:

$$\begin{aligned}&\frac{d}{dt}\mu _{v}\left( q_{v},t\right) =-\frac{d}{dr_{i^{*}\left( q_{v}\right) }\left( q_{v}\right) }\mu _{v}\left( q_{v},t\right) \nonumber \\&\quad +\delta \left( 0,\tau \left( e\left( q_{v}\right) \right) \right) \mu _{v}\left( q_{v}\ominus e\left( q_{v}\right) \right) \left[ \sigma _{tr}\left( q_{v}\ominus e\left( q_{v}\right) ,q_{v}\right) +\sigma _{e}\left( q_{v}\ominus e\left( q_{v}\right) ,q_{v}\right) \right] \nonumber \\&\quad -\mu _{v}\left( q_{v},t\right) \sum _{q_{v}^{\prime }}\left[ \sigma _{tr}\left( q_{v},q_{v}^{\prime }\right) +\sigma _{e}\left( q_{v} ,q_{v}^{\prime }\right) \right] \nonumber \\&\quad +\int _{q_{v}^{\prime }:q_{v}^{\prime }\ominus C\left( q_{v}^{\prime }\right) =q_{v}}d\mu _{v}\left( q_{v}^{\prime }\right) \sigma _{f}\left( q_{v}^{\prime },q_{v}^{\prime }\ominus C\left( q_{v}^{\prime }\right) \right) \nonumber \\&\quad -\mu _{v}\left( q_{v}\right) \sigma _{f}\left( q_{v},q_{v}\ominus C\left( q_{v}\right) \right) +\sum _{v^{\prime }\text {n.n.}v}\beta _{vv^{\prime } }\left[ \mu _{v^{\prime }}\left( q_{v}\right) -\mu _{v}\left( q_{v}\right) \right] . \end{aligned}$$

(30)

Compared to the setting of [

3], we have the following simplifications:

- 1.
The graph *G* is the lattice \(\mathbb {Z}^{1};\)

- 2.
All customers have the same class;

- 3.
The service time distribution \(\eta \) is exponential, with the mean value 1;

- 4.
The service discipline is FIFO;

- 5.
The exogenous customer *c* arriving to node *v* has for destination the same node *v*; inflow rates at all nodes are equal to \(\lambda ;\)

- 6.
The two servers at \(v,v^{\prime },\) which are neighbors in \(\mathbb {Z} ^{1}\) exchange their positions with the same rate \(\beta \equiv \beta _{vv^{\prime }};\)

The equation for the fixed point then becomes:

$$\begin{aligned} 0=&\mu _{v}\left( q_{v}\ominus e\left( q_{v}\right) \right) \left[ \sigma _{tr}\left( q_{v}\ominus e\left( q_{v}\right) ,q_{v}\right) +\sigma _{e}\left( q_{v}\ominus e\left( q_{v}\right) ,q_{v}\right) \right] \\&-\mu _{v}\left( q_{v}\right) \sum _{q_{v}^{\prime }}\left[ \sigma _{tr}\left( q_{v},q_{v}^{\prime }\right) +\lambda \right] + \sum _{q_{v}^{\prime }:q_{v}^{\prime }\ominus C\left( q_{v}^{\prime }\right) =q_{v}}\mu _{v}\left( q_{v}^{\prime }\right) \\&-\mu _{v}\left( q_{v}\right) \mathbb {I}_{q_{v}\ne \varnothing } +\sum _{v^{\prime }=v\pm 1}\beta \left[ \mu _{v^{\prime }}\left( q_{v}\right) -\mu _{v}\left( q_{v}\right) \right] ~. \end{aligned}$$

The proof is concluded when using the fact that queue

\(q_{v}\) can in this setting be identified with the sequence of destinations

\(D(c_{i}) \) of its customers.