Travelling Wave Solutions to Multiserver Queue and Gossiping Secretaries Problems

N=1 to 10, t=0 to 500

The evolution of the density profiles for the multiserver network in the case N=1 to 10.

Density profiles for the multiple servers network.

The network is described in section 4 of our paper. Here we consider the special case of totally asymmetric p = 1 and the initial condition (41) where u_l = 1 and u_r = 0; other situations with Riemann-type data can treated in a similar fashion and will be analyzed elsewhere. So, from now on in this subsection

H(u) =

é
ë

æ
è

1-u

ö
ø

æ
è

1-u

ö
ø

N+1

ù
û

Thus, for x such that v¢(x) ¹ 0, the equation (46) can be written in the form

-1+(N+1)(1-v)^N = x,

so that the solution of the Riemann problem is

u(t,x) =

ì
ï
ï
í
ï
ï
î

for x < -t;

1-[(x/t+1)/(N+1)]^1/N,

for -t £ x < Nt;

for Nt £ x.

N=1, t=0 to 500

The evolution of the density profiles for the multiserver network in the case N=1. This is the network considered by Benassi and Fouque (1987) and the Burgers equation in the hydrodynamic limit.

N=4, t=0 to 500

The evolution of the density profiles for the multiserver network in the case N=4.

N=100, t=0 to 500

The evolution of the density profiles for the multiserver network in the case N=100.

Density profiles for the GS network.

This case is slightly more difficult as uniqueness questions arise because of the bifurcations. For the GS network, we examine the solution with initial condition u_l = 2/(m+1), u_r = 0, that is, we begin with an average of m-1 customers for every 2 servers to the left of zero and no servers to the right of zero. Proceeding as in the multiserver case, we have

H(u) = u(1-u)^m,

(1)

so for x such that v(x) ¹ 0, the equation (46) can be written in the form

(1-u)^m-1(1-(m+1)u) = x.

(2)

So, for the GS network the solution of the Riemann problem bifurcates at (x_b,u_b), where

x_b

-t

æ
ç
è

m-1

m+1

ö
÷
ø

m-1

(3)

u_b

2/(m+1).

(4)

The figure below illustrates the case where m = 4, but the solutions are similar to those shown for m > 1. When m = 1 we have a simple asymmetric exclusion process.

m=4, t=0 to 20

The evolution of the density profiles for the gossiping secretaries network in the case m=4. The graph shows a bifurcation. The lower branch corresponds to the entropy solution.

N=m=4, t=0 to 1200

The evolution of the density profiles for the gossiping secretaries and multiserver networks in the case N=m=4.