Some Open Problems Related to Stability

Some open problems related to stability S. Foss Heriot-Watt University, Edinburgh and Sobolev’s Institute of Mathematics, Novosibirsk I will speak about a number of open problems in queueing. Some of them are known for decades, some are more recent. They relate to stability and to rare events. There is an idea to prepare a special issue of QUESTA on open problems, and this text may be considered as a prospective contribution to that. The choice of open problems reflects the speaker’s own interests, and should not be taken as suggesting that these are the only, or even most important, problems! 1 Multi-server queue with First-Come-First-Served discipline A system with a finite number of identical servers and with FCFS service discipline is one of the simplest models in queueing theory. It has been known for a long time. To the best of my knowledge, Kiefer and Wolfowitz were the first who studied it rigorously. 1.1 Convergence in the total variation to the stationarity Consider first a single-server first-come-first-served (FCFS) queue G/G/1 with interarrival times {tn } between the arrivals of the nth and (n + 1)st customers, and service times of the nth customer {σn }. Assume that the two-dimensional sequence {tn , σn } is stationary ergodic and that the queue is stable, that is Eσ1 < Et1 . Let Wn , n ≥ 1 be the waiting time of customer n in the system (before the start of its service). Then W1 = x ≥ 0, and Wn+1 = max(0, Wn + σn − tn ) ≡ (Wn + ξn )+ , n≥1 where x is the initial delay, ξn = σn − tn , and x+ = max(0, x). P Let S0 = 0 and Sn = n1 ξi , n ≥ 1. Clearly, Wn = max(0, x + Sn , Sn − S1 , Sn − S2 , . . . , Sn − Sn−1 ) which implies that there exists a proper limiting distribution which does not depend on x. In other words, there exists a unique stationary distribution of the waiting time and, for any initial delay, there is a convergence to stationarity, and the convergence is in the total variation norm. There are many known ways to establish this result. The best one seems to use the “Loynes scheme”. Without loss of generality, we may assume (σn , tn ) to be defined for all −∞ < P n < ∞. Let Se0 = 0 and Sen = nj=1 ξ−j , n ≥ 1. Then Wn =st Mn(x) := max(0, x + Sen , Sen−1 , . . . , Se1 ). 1 (1) Denote M = supn≥0 Sen . From the SLLN, Sen → −∞ a.s., so M < ∞ a.s. Moreover, the time ν ≡ ν (x) = max{n ≥ 0 : x + Sen ≥ 0} is finite a.s. and, therefore, Mn(x) = M for all n > ν (x) . So, M is the unique limiting distribution and the convergence in total variation follows from the coupling inequality: for any x ≥ 0, sup |P(Wn ∈ A)−P(M ∈ A)| = sup |P(Mn(x) ∈ A)−P(M ∈ A)| ≤ P(ν (x) > n) → 0, A n → ∞. A (0) In particular, if x = 0, then – as follows from (1) – the sequence Mn := Mn , n ≥ 1 is monotone increasing a.s. and couples with M , starting from time ν (0) + 1. Now let m > 1 be a positive integer and consider the G/G/m FCFS queue with interarrival times {tn } and service times {σn }. Assume again that the two-dimensional sequence {tn , σn } is stationary ergodic and that the system is stable. Here the stability means Eσ1 < mEt1 . Consider Kiefer-Wolfowitz vectors of virtual waiting times Wn = (Wn1 , . . . , Wnm ) which satisfy the recursion W1 = x ≥ 0 and Wn+1 = R(Wn + e1 σn − 1tn )+ , n≥1 where x is a vector of initial delays, e1 = (1, 0, . . . , 0) is a unit vector, 1 = (1, 1, . . . , 1) is a vector of units, and operator R rearranges coordinates of a vector in weak ascending order. In particular, Wn,1 is a waiting time of customer n. It is known that, in general, in a multi-server queue there may be many stationary regimes [20]; there exist the minimal and the maximal stationary distribution [2, 3], and there are some relations between stationary distributions [7]. Similarly to the one-dimensional case, (x) one can define vectors Mn which satisfy a recursion which is more complex than (1). In (0) particular, if there are no initial delays, x = 0, then the vectors Mn = Mn are monotone weakly ascending (coordinate-wise and a.s.) and converge a.s. to a limiting random vector which has the minimal stationary distribution πmin (see e.g. [25]). But this implies only the weak convergence of the distributions of these vectors, and not the convergence in the total variation. If the sequence {(σn , tn )} satisfies in addition some good “mixing” properties (say, is i.i.d. or regenerative), then one can again show the uniqueness of the stationary regime and the convergence in the total variation starting from each initial value, using either Harris properties (in the Markovian case, see e.g. [21]) or the renovation techniques (in a more general setting, see e.g. [4]). The conjecture is: assuming only that the sequence {(σn , tn )} is stationary ergodic, that the stability condition holds, and that the initial value is 0, then there is no need for any further restriction to establish the convergence in the total variation: sup |P(Wn ∈ A) − π(A)| → 0, n → ∞. A There have been a number of unsuccessful attempts to prove this conjecture (see, e.g., [23]). 2 1.2 Existence of Moments Assume now that {σn } and {tn } are two i.i.d. sequences that do not depend on each other. Continue to assume the stability condition ρ := Eσ1 /Et1 < m holds. Recall that in this case the stationary distribution is unique. Denote by D the stationary waiting time in the multi-server queue. Fix γ > 0 and formulate the following question: what are the conditions for EDγ to be finite. A correct (but partial!) answer to this question has been obtained recently by SchellerWolf and Vesilo [26]. To be completely exact, the result below was not formulated by these authors but may be deduced from their results. Denote by BI the integrated service time distribution,   Z ∞ P(σ1 > y)dy . BI (x) = 1 − min 1, x Let σI,1 , . . . , σI,m be i.i.d. random variables with common distribution BI . Proposition Assume that ρ is not an integer and denote by k ∈ {0, 1, . . . , m − 1} its integer part. Then EDγ < ∞ iff E (min(σI,1 , . . . , σI,m−k ))γ < ∞. The proofs of the results in [26] are based on the construction of an auxiliary, so-called “semi-cyclic” service discipline. A direct proof of the proposition may be found in [13] and is based on ideas close to ideas of Keifer and Wolfowitz [16]. An open problem here is: what are the conditions for existence (finiteness) of power moments of D if ρ is an integer. 1.3 Rare events Assume again that {σn } and {tn } are two i.i.d. sequences that do not depend on each other. Continue to assume the stability condition ρ := Eσ1 /Et1 < m to hold. Again let D be the stationary waiting time. We formulate the following questions: what may the asymptotics for P(D > x) be when x is large and what is the “typical” sample path which lead to such a large value of the stationary waiting time. To answer (only partially!) these questions, we need further restrictions on the distribution of service times. First, we assume that the common distribution of service times is heavy-tailed, i.e. for any c > 0, its cth exponential moment does not exist, Eecσ1 = ∞. Whitt [27] formulated the following conjecture: if k ≤ ρ < k + 1 for an integer k < m, then
m−k P(D > x) ∼ γ B I (ηx) as x → ∞ (2) “where γ and η are positive constants (as functions of x)” [sic, [27]] and where B I is the tail of the integrated service time distribution. Intuitively, formula (2) says that the main cause for the stationary waiting time to be large is to have m − k big service times in the past. In [12] and [13] we show that if the distribution of service times is intermediate regularly varying and if ρ is not an integer, then the conjecture of Whitt is correct with γ = γ(x) squeezed between two positive constants and with η being a constant. Also, the conjecture holds if ρ ∈ (0, 1) and if the distribution BI is any subexponential distribution. Also, we 3 found that if m = 2, ρ < 2, ρ 6= 1, and the service times distribution is again intermediately regularly varying, then γ is a constant. The open problems here are (I formulate them in the particular case of a two server queue, m = 2): find the asymptotics for P(D > x) (i) if ρ = 1 – at least, for some particular subexponential (say, regularly varying) distribution of service times; (ii) if ρ ∈ (1, 2) and if the distribution of service times is heavy-tailed but has all power β moments finite, for example, if P(σ1 > x) = e−x , for some β ∈ (0, 1). It would be great to understand what are in these cases the “typical” paths which lead to large values of D. 2 Further problems on multi-server queues Consider again the multi-server (say, 2-server) queue with stationary and ergodic input {tn , σn }, but assume now that the discipline is “join-the-shortest-queue”: there are individual queues in front of the servers, and each arriving customer joins immediately the shortest queue (or one of the shortest at random if there are many). So, here are the open problems: – how many stationary regimes may exist if we do not assume any extra condition in addition to the obvious stability condition? – what are the minimal requirements for the uniqueness of the stationary regime? – under what conditions (none?) do we have weak (or TV) convergence? The model exhibits NO monotonicity. It is not amenable to Loynes-type schemes. It is entirely open ([18]). 3 Greedy service mechanism There are many circumstances in our life where we may ask the question, is a “locally optimal” (“greedy”) mechanisms also optimal in the long-run? Below are examples of mathematical models where the answer to such a question is open. There are two continuous state space models where the stability conjecture is obvious, but nobody is able to verify it. In both models, the driving algorithm contains a “locally optimal” (“greedy”) element. It looks like none of the existing stability methods works here. 3.1 Stability of a greedy server A single server is located on the circle. Particles arrive in a Poisson stream of rate λ and are uniformly distributed (as material points) on the circle (people say that there is a “Poisson rain” of particles). It takes a single unit of time to serve a particle. After any service, the particle disappears, the server chooses to serve next the closest particle and moves to it with a (positive finite) constant speed (ignoring new arrivals), serves it during another unit of time, then chooses the next closest particle and moves to it, etc. The conjecture is: this model is stable for any λ < 1. A plausible “proof” might be as follows: if the number of requests is very large, then the server is busy with service almost all the time (with a service speed close to one), and then we may apply, say, fluid approximation ideas to deduce the stability. 4 This model and this conjecture have already been known for more than 20 years, see [5], but nobody has been able to succeed with obtaining either a proof or a counter-example here. The key problem is the continuity of the state space, and there are several results (see, e.g., [9, 10, 24] for further details) with the proof of a similar hypothesis for models with a finite state space (for instance, you may replace the continuous circle by a finite lattice on it). If the server uses any “state-independent” algorithm for moving (say, always walks in the left direction or chooses the next direction with probability 1/2 independently of everything else), then it is easy to verify the conjecture using the ideas explained above – see, e.g., [6, 19]. 3.2 Stability of a model with two streams: stream of customers and stream of servers Again, there is a circle, but this time no server or service. Instead, there are two independent Poisson streams/rains, of “black” and of “white” particles, with rates λ and 1, respectively. Black particles arrive at the circle and stop there, but white particles pass straight through the circle (this means they “arrive and immediately disappear”). There is given a distance ε > 0. When a white particle passes through the circle at some point, it observes all blacks in the ε-neighbourhood and takes (deletes) the one which is the closest to itself (if there are any black particles at that instant). The natural conjecture is: stability should be guaranteed by the condition λ < 1, independently of the circle length and the number ε. But the problem is open too. Again, there exist simple proofs for stability if the model is modified: if either the continuous state space (the circle) is replaces by a finite set, or the greedy mechanism is replaced by any state-independent mechanism (for instance, if a white particle takes one of blacks from the neighbourhood “at random”, with equal probabilities) (see [1]). 4 Stability may depend on the whole distribution The conjecture is: even in simple queueing systems, the stability conditions may depend both on the initial values and on the whole distribution of the driving sequences. Here is an example where the conjecture may be true (see [8] for more detail). Consider a system with three servers (numbered 1 to 3) fed by a Poisson process with intensity λ. There are three classes of customers and each arriving customer becomes a class i customer (i = 1, 2, 3) with probability 1/3. Class 1 customers may be served by 1st and 2nd servers (where 1st server is “left” and 2nd server is “right”), class 2 customers by 2nd and 3rd servers (here 2nd server is “left” and 3rd is “right”), and class 3 customers by 3d and 1st servers (here 3rd is “left” and 1st is “right”). Upon arrival, a customer chooses an accessible server with the shorter workload. There are two probability distributions, Fl and Fr , and a customer’s service time has distribution Fl if it is served by its left server and Fr otherwise. Simulations show that the conjecture may hold, but there is no a rigorous proof. 5 5 Random fluid limits and positive Lebesgue measure of the area of null-recurrence Consider an open polling system with two stations and two “heterogeneous” servers. Each station i = 1, 2 has a Poisson input with intensity λi = 1. For i, j ∈ {1, 2}, service times (j) of server j at station i are i.i.d. exponential with intensity µi . Both servers follow the so-called exhaustive service policy: after completing a service, a server either starts with a service of a new customer (if there is any), or leaves the station for the other one. We assume for simplicity that “walking” (“switchover”) times are equal to zero. If there is no free customer at either station, the server becomes “passive”. The system described has a nice fluid model where all fluid limits are random and piecewise (j) deterministic. The system is characterised by 4 parameters {µi }i,j=1,2 , and all of them have to be less than one in order to make the system stable (but this is definitely not sufficient for stability). (j) The conjecture here is: in the positive 4-dimensional cube, the set of parameters {µi } for which the system is “null-recurrent” has a positive Lebesgue measure. See [14] for more details. 6 Multi-access channel with protocols based on partial information Consider a single channel which is shared among many users and transmits packets (messages) of a single length. Assume that time is slotted and each service time is equal to the slot length. The number of packages arriving into the system during a time slot is Poisson with parameter λ. At the beginning of time slot n, each package is trying to be transmitted with the same probability pn independently of everything else. If two or more packages try to transmit simultaneously, the transmissions collide and packages stay in the system and have to try again later. If there is only one transmission, then it is successful and the package leaves the system. If there are no transmissions, then the slot is empty. Denote by Wn the number of packages in the system at the beginning of nth time slot. Assume that probabilities pn are defined inductively and that pn+1 may depend only on pn and the “binary” information of whether there is a successful transmission in slot n or not. Then the pairs (Wn , pn ) form a time homogeneous Markov chain. The open question is: in the class of procedures (protocols) described above, does there exist a protocol which makes the Markov chain (Wn , pn ) positive recurrent (ergodic), for some positive λ? See [15] for more detail. It is known [17, 22] that if a choice of pn+1 is based on pn and of either of two other “binary” informations (nth slot was empty or not; or there was collision in nth slot or not), then λ < e−1 is necessary and sufficient for the existence of an ergodic protocol. References 1. V. Anantharam, private communication. 2. A. Brandt. On stationary waiting times and limiting behaviour of queues with many servers II. The G/GI/m/∞ case. Elektron. Informationsverarb. Kybernetik, 21 (1985), 151–162. 3. A. Brandt, P. Franken and B. 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