Quasi-stationary distributions for queueing and other models

We will examine questions concerning quasi-stationary behaviour in evanescent processes. The idea has its origins in biological modelling, where typically we are interested in limiting behaviour conditional on non-extinction. For queueing processes we are typically interested in the behaviour within a busy period in stable or near stable queues Kij93a ; Kyp72b , or, for unstable queues, prior to last exit from the empty state CHP00 .

Let $(X(t),\,t\geq 0)$ be a Markov chain in continuous time whose state space $S=\{0\}\cup C$ consists of transient states $C=\{1,2,\dots\}$ and an absorbing state 0. Let $Q=(q_{ij},\,i,j\in S)$ be the $q$-matrix of transition rates, assumed to be stable, conservative and regular, so that there is a unique transition function $P(t)=(p_{ij}(t),\,i,j\in S)$ associated with $Q$, and $p_{ij}(t)=\Pr(X(t)=j|X(0)=i)$. We assume that $C$ is irreducible, and that $0$ is reached with probability $1$ from any state in $C$. Thus, in particular, for $i,j\in C$, $p_{ij}(t)\to 0$ as $t\to\infty$, and, for $i\in C$, $p_{i0}(t)\to 1$. We are interested in the limit of the ratio

called a limiting conditional distribution (LCD). The first general results on LCDs SV66 were facilitated by a finer classification of transient states Kin63a and a common rate at which the transition probabilities decay: there is a $\lambda\geq 0$, called the decay parameter (of $C$), such that $t^{-1}\log p_{ij}(t)\to-\lambda$ as $t\to\infty$, for all $i,j\in C$. $C$ is then $\lambda$-recurrent or $\lambda$-transient according to whether $\int_{0}^{\infty}e^{\lambda t}p_{ij}(t)\,dt$ diverges or converges (for some, and then all, $i,j\in C)$, and, when $C$ is $\lambda$-recurrent, it is $\lambda$-positive or $\lambda$-null according to whether the limit $\lim_{t\to\infty}e^{\lambda t}p_{ij}(t)$ is positive or zero (for some, and then all, $i,j\in C)$. When positive, its value is determined by $(m_{i},\,i\in C)$ and $(x_{i},\,i\in C)$ satisfying

Unique positive solutions to (2) (called the $\lambda$-invariant measure and vector, respectively) are guaranteed when $C$ is $\lambda$-recurrent. $C$ is then $\lambda$-positive if and only if $A^{-1}:=\sum_{k\in C}m_{k}x_{k}<\infty$, whence $\lim_{t\to\infty}e^{\lambda t}p_{ij}(t)=Ax_{i}m_{j}$. So, one can see, at least formally from (1), that, since $p_{i0}(t)=1-\sum_{j\in C}p_{ij}(t)$,

$\lambda$-positivity is indeed sufficient Ver69 , the limit taken to be $0$ when $\sum_{k\in C}m_{k}=\infty$.

One might think this completes the picture. However, $\lambda$-positivity is not necessary for the existence of an LCD (see the example below). Further, the decay parameter cannot usually be determined from $Q$, and $\lambda$-positivity cannot usually be checked from $Q$. Of course $p_{ij}(t)$ is seldom available explicitly, but there are “$q$-matrix versions” of (2),

and positive solutions to (3) satisfy (2) under conditions that are easy to check Pol86 .

One way to approach the question of whether LCDs exist for $\lambda$-transient chains is to characterise the smallest $\kappa>1$ such that $t^{\kappa}e^{\lambda t}p_{ij}(t)$ has a strictly positive limit for all $i,j\in C$. Such a characterisation is presently unavailable. I conjecture (for $\lambda$-null and $\lambda$-transient chains) that (i) when such a $\kappa$ exists, it is the same for all $i,j\in C$, that (ii) the limit is always of the form $Ax_{i}m_{j}$, where $(m_{i},\,i\in C)$ and $(x_{i},\,i\in C)$ satisfy (2) perhaps with an inequality ($\leq$), and (iii) there is a $\kappa_{0}\leq\kappa$ such that $t^{\kappa_{0}}e^{\lambda t}(1-p_{i0}(t))\to Bx_{i}$.

The approach proposed here contrasts with work on discrete-time chains, which highlight the many complications in the theory of $R$-transient chains McDF17 ; Kes95 , and more in line with work on algebraic transience MS14 and sub-geometric convergence for ergodic discrete-time Markov chains DMS07 (which has become important in the analysis of Markov chain Monte Carlo methods). The condition that $\kappa$ exists is equivalent to requiring that the function $g_{ij}(t):=e^{\lambda t}p_{ij}(t)$ is regularly varying with index $\alpha=\kappa^{-1}$ (the same index of regular variation for all $i$ and $j$), or, equivalently, that there is a $\kappa>1$, the same for all $i,j\in C$, such that $t^{\kappa}g_{ij}(t)$ is slowly varying Sen76 . Conjectures (i) and (ii) are true for $\lambda$-null recurrent chains (Lemma 1 of Pol01 ), and indeed (iii) with $\kappa_{0}=\kappa$ under an additional condition. Critical to the argument is that the $\lambda$-subinvariant measures and vectors ((2) with the inequality) are unique and $\lambda$-invariant. This is not true in the $\lambda$-transient case. One might hope to adapt Kingman’s arguments based on inequalities derived from the Chapman-Kolmorogov equations Kin63a , but I cannot see how. In addition to the example detailed above, the conjectures are supported by several other contrasting models: the M/M/1 queue with $p>q$, the random walk on $\mathbb{Z}$ in continuous time Kin63a , the birth-death immigration process And91 , quasi-birth-death processes BBLPPT97 , and various branching models AH83 . Interestingly, the critical Markov branching process provides an example for which $\kappa=2$ and $\kappa_{0}=1$.