Perfect sampling of GI/GI/c queues

Abstract

We introduce the first class of perfect sampling algorithms for the steady-state distribution of multi-server queues with general interarrival time and service time distributions. Our algorithm is built on the classical dominated coupling from the past protocol. In particular, we use a coupled multi-server vacation system as the upper bound process and develop an algorithm to simulate the vacation system backward in time from stationarity at time zero. The algorithm has finite expected termination time with mild moment assumptions on the interarrival time and service time distributions.

Appendix: A list of selected notation

Notation	Meaning
$A_n:n\in {\mathbb {Z}}\backslash \{0\}$	Interarrival time between the arrivals at $T^0_n$ and $T^{0,+}_n$ (Sect. 2.1)
$D_n^0:n\in {\mathbb {Z}}\backslash \{0\}$	Waiting time of the customer arriving at $T^0_n$ in the vacation system (Sect. 2.1.2)
$E_u(t;z):t\ge 0$	Time elapsed since previous arrival at time $u+t$ when the time elapsed since previous arrival at time u is the corresponding subvector of z, i.e., e (Sect. 2.2.1)
$i(n):n\in {\mathbb {Z}}\backslash \{0\}$	Label of the server who serves the customer arriving at $T^0_n$ (Sect. 2.1.2)
$M(t):t\ge 0$	Running time maximum of process $\{X(s):s\ge t\}$ (Sect. 4)
$N^0_u(t):t\ge 0$	Number of arrivals during $[u,u+t]$ (Sect. 2.1)
$N^0(t)$	Number of arrivals during [0, t] if $t\ge 0$ or in [t, 0] if $t<0$ (Sect. 2.1)
$N^i_u(t):t\ge 0$	Number of activities initiated by server i during $[u,u+t]$ (Sect. 2.1)
$N^i(t)$	Number of activities initiated by server i during [0, t] if $t\ge 0$ or [t, 0] if $t<0$ (Sect. 2.1)
$Q_u(t;z):t\ge 0$	Number of people waiting in queue at time $u+t$ of a GI/GI/c queue that starts at time u and is initialized with z (Sect. 2.2.1)
$Q_v(t)$	Number of people waiting in queue in stationary vacation system at time t (Sect. 2.1.1)
$R_u(t;z):t\ge 0$	Ordered (non-decreasing) remaining service times of all c servers at time $u+t$ of a GI/GI/c queue that starts at time u and is initialized with z (Sect. 2.2.1)
$\{S^{(0)}_n:n\ge 0\}$	Random walk with negative drift associated with the arrival renewal process (Sect. 4)
$\{S^{(i)}_n:n\ge 0\}$	Random walk with negative drift associated with the activity renewal process of server i (Sect. 4)
$T^0_n:n\in {\mathbb {Z}}\backslash \{0\}$	n-th ($(-n)$-th) arrival time counting forward (backward) in time from time 0 (Sect. 2.1)
$T^{0,+}_n:n\in {\mathbb {Z}}\backslash \{0\}$	Next arrival time after $T^0_n$ (Sect. 2.1)
$T^{0,-}_n:n\in {\mathbb {Z}}\backslash \{0\}$	Previous arrival time before $T^0_n$ (Sect. 2.1)
$T^i_n:n\in {\mathbb {Z}}\backslash \{0\}$	n-th ($(-n)$-th) activity initiation time of server i counting forward (backward) in time from time 0 (Sect. 2.1)
$T^{i,+}_n:n\in {\mathbb {Z}}\backslash \{0\}$	Next activity initiation time of server i after $T^i_n$ (Sect. 2.1)
$T^{i,-}_n:n\ge {\mathbb {Z}}\backslash \{0\}$	Previous activity initiation time of server i before $T^i_n$ (Sect. 2.1)
$U^0(t)$	Time until next arrival from time t (Sect. 2.2.2)
$U^i(t)$	Time until next activity initiated by server i from time t (Sect. 2.2.2)
U(t)	c-dimensional vector $\left( U^1(t),\ldots ,U^c(t)\right) ^T$ (Sect. 2.2.2)
$V_n:n\in {\mathbb {Z}}\backslash \{0\}$	Service time of the customer who arrives at $T^0_n$ (Sect. 2.1.1)
$V^i_n:n\in {\mathbb {Z}}\backslash \{0\}$	Length of the activity of server i that is initiated at $T^i_n$ (Sect. 2.1)
$W(n):n\in {\mathbb {Z}}\backslash \{0\}$	Kiefer–Wolfowitz workload vector at time $T^0_n$ of a stationary GI/GI/c queue (Sect. 2.2.1)
$W_k\left( T_n^0;w\right)$: $n\ge k$ and $n,k\in {\mathbb {Z}}\backslash \{0\}$	Kiefer–Wolfowitz workload vector at time $T^0_n$ of a GI/GI/c queue which has its workload vector at time $T^0_k$ being w (Sect. 2.2.1)
$W_v(n):n\in {\mathbb {Z}}\backslash \{0\}$	Analog Kiefer–Wolfowitz workload vector at time $T^0_n$ of the stationary vacation system (Sect. 2.2.2)
$W_v(n+):n\in {\mathbb {Z}}\backslash \{0\}$	Analog Kiefer–Wolfowitz workload vector at time $T^{0,+}_n$ of the stationary vacation system (Sect. 2.2.2)
X(t)	$X_0(t)$ if $t\ge 0$ or $X_t(-t)$ if $t<0$ (Sect. 2.1.1)

$X_u(t):t\ge 0$	$N^0_u(t)-\sum _{i=1}^cN^i_u(t)$ (Sect. 2.1.1)
Z(t)	State vector of a stationary GI/GI/c queue at time t
$Z_u(t;z):t\ge 0$	State vector at time $u+t$ of a GI/GI/c queue that starts at time u and is initialized with z (Sect. 2.2.1)
$\sigma ^i_u(t):t\ge 0$	Number of service initiations by server i during $[u,u+t]$ (Sect. 2.1.2)
$\varPhi _j^i:j\ge 0,0\le i\le c$	j-th downward “milestone” of random walk $\{S^{(i)}_n:n\ge 0\}$ (Sect. 4)
$\varUpsilon _j^i:j\ge 0,0\le i\le c$	j-th upward “milestone” of random walk $\{S^{(i)}_n:n\ge 0\}$ (Sect. 4)