Stability and Tail Asymptotics in a Multiclass Queue with State ...

arXiv:1609.03999v1 [math.PR] 13 Sep 2016

Stability and Tail Asymptotics in a Multiclass Queue with State Dependent Arrival Rates- First Draft S. Asmussen, P. Ernst, J. Hasenbein

1

Introduction

We study a single server multiclass queueing system in which the arrival rates depends on the current job in service. Thus, instead of a vector of arrival rates, the system is characterized by a matrix of arrival rates. This model differs from previous state-dependent models, in which the parameters depend primarily on the number of jobs in the system, rather than the job in service. We first obtain necessary and sufficient stability conditions via fluid models. Then, making a connection with multitype Galton-Watson processes, we characterize the Laplace transform of busy periods in the system.

2

The Queuing Model

We consider a multiclass single-server queue described as follows. There are K classes of jobs, each arriving according to independent counting processes, whose arrival rate depends on the class of the job in service. Namely, if the server is serving a job of class j, the arrival rate of class-i jobs is λij , i, j = 1, . . . , K. We then define the matrix of arrival rates to be Λ = (λij ), i, j = 1, . . . , K. Furthermore, if no job is in service, then the arrival rate of class-i jobs is λi0 , i = 1, . . . , K. We describe the arrival mechanism more precisely in the dynamical equations later in this section. ¯ j = PN λij for each j = 1, . . . , K. Service times for class-i jobs are i.i.d. with a general Let λ i=1 distribution function Fi , i = 1, . . . , K. Let Si be a generic service time for class-i jobs, with E[Si ] = mi = µ−1 i , i = 1, . . . , K. Let G = diag(µ1 , µ2 , . . . , µK ). We can then form the “mean offspring matrix” M = ΛG−1 where the terminology refers to the ijth element λij mj giving the mean number of arriving ∗ class j customers during service of a class i customer. Finally, let Fi be the cumulative distribution function of Si , i = 1, . . . , K. Let ψi beR the Laplace-Stieltjes transform (LST) of Si , i = 1, . . . , N , ∞ respectively, that is, ψi (s) = E[e−sSi ] = 0 e−st dFj (s) for s > 0. The service disciplines we consider are non-idling, i.e., jobs must be served using the full capacity of the server, whenever there are jobs in the system. For now, we only consider disciplines in which one job can be served at a time by the server (e.g., FCFS, LCFS, priority disciplines). The model we formulate actually allows for any head of the line (HL) discipline, in which the server may serve multiple jobs simultaneously, but at most one job from each class may be served. Since we are only concerned with the distribution of the busy period, we need not specify the order in which the ∗ customers are served (we could even allow for disciplines like shortest job first!). Let Qi denote the steady-state number of class-i jobs in the system, i = 1, . . . , K, and set Q = (Q1 , . . . , QK ). Each state of the system takes nonnegative integer values, that is, x = (x1 , . . . , xK ) ∈ K ZK + . Let Bx be the busy period when the system starts from the state x ∈ Z+ , that is, the time period ∗ 1

until the system becomes empty. When x consists of a single customer of class i, we just write Bi Let gx be the LST of Bx , i.e., gx (s) = Ex [e−sBx ] for x ∈ ZK + and s > 0. Let B denote the busy period for ∗ the system starting from the time epoch of an arbitrary new arrival with an empty system until the system empties next.

3 3.1

The Corresponding Fluid Model Dynamical Equations

We now more formally define the queueing model. For i ∈ {1, . . . , K} and t ≥ 0, let Qi (t) denote the number of class i jobs in the system at time t, whether in service or in the queue. Similarly, let Ti (t) denote the amount of time that has been devoted to serving class i jobs in [0, t]. We also define Ai (t) and Di (t) which are, respectively, the total number of class i jobs that have arrived and departed from the system in [0, t]. We then have the following input-output equation for each job class i: Qi (t) = Qi (0) + Ai (t) − Di (t).

(1)

For each class i, the counting process Eij (t) be the number of class i tracks the number of arrivals of class i customers during the first t time units devoted to processing class j. E 0 (t) counts the number of class i arrivals during the first t time units that no job is being processed at the server. Then the total number of class i arrivals in [0, t] is given by Ai (t) = Ei0 (T0 (t)) +

N X

Eij (Tj (t)).

(2)

j=1

As mentioned above, the processes Eij for 0 ≤ i, j ≤ K are assumed to be mutually independent Poisson processes. As for the service processes, for each i, 1 ≤ i ≤ K and positive integer n, we let Vi (n) denote the total service requirement for the first n class i jobs. Assuming an HL service discipline, we have the usual relations: Vi (Di (t)) ≤ Ti (t) ≤ Vi (Di (t) + 1) (3) for each t ≥ 0 and 1 ≤ i ≤ N .

3.2

Fluid Model

For purposes of determining the stability conditions of a more general version of our model, we formulate a fluid network version of the model. For references to important definitions and results in the fluid model literature, we refer to the survey of [6] and to the textbook of [3]. For i ∈ {1, . . . , K} and t ≥ 0, let Qi (t) denote the amount of fluid of class i in the system at time t. Similarly, let Ti (t) denote the amount of time that has been devoted to serving class i fluid in [0, t]. We also define Ai (t) and Di (t) which are, respectively, the total amount of class i fluid that has arrived and departed from the system in [0, t]. Then it is clear that we have the following standard equation: Qi (t) = Qi (0) + Ai (t) − Di (t), (4) for each i ∈ {1, . . . , K} and t ≥ 0. The departure processes in this system also obey the standard relation Di (t) = µi Ti (t). 2

The unusual feature of this model lies in the arrival process, which is dependent on the current class in service. In the queueing model, processor sharing is not allowed. Hence, there is (at most) one class in service at any given time and “the customer in service” is defined unambiguously. Here, we provide a more general formulation that reduces to the queueing model presented in earlier sections, under appropriate restrictions on the allowable queueing disciplines. First, we recall the usual condition N X

T˙i (t) ≤ 1,

(5)

i=1

which simply indicates that the server cannot devote more than 100% of its time to serving fluids of P ˙i (t) = 1 all classes. In fact, we will assume that the queueing discipline is non-idling and thus N T i=1 whenever there is a positive amount of fluid in the system. We also define the idle time in [0, t] to be: Y (t) = t −

N X

Ti (t).

i=1

Next, note that the current arrival rate of class i fluid is given by A˙ i (t). In the queueing model, if a job of class j is in service then the arrival rate of class i jobs is λij . Let λi be the column vector ′ ′ ˙ ˙ ˙ (λ1i , . . . λN i ) and let T(t) be the column vector (T1 (t), . . . , TN (t)) . Then, define the fluid arrival rate of class i to be: ˙ A˙ i (t) = λi0 · Y˙ (t) + λ′i · T(t), (6) In particular, when there is fluid in the system, the class i arrival rate is a convex combination of the elements of λi . If we restrict to policies in which only one class can be served at any time, then (6) assigns an arrival rate of λij to class i fluid when class j fluid is in service. This concurs with the queueing model. Putting all of this together we then have: Z t ˙ (λ0i · Y˙ (t) + λ′i · T(u)) du − µi Ti (t) (7) Qi (t) = Qi (0) + 0

= Qi (0) + λi0 · Y (t) + λ′i · T(t) − µi Ti (t).

(8)

Writing this in matrix form yields Q(t) = (M − I)D(t) + Y (t) · λ0 .

4

(9)

Stability Results for Fluid Model

Let f (Q(t)) = eG−1 Q(t) be the workload function. Note that Ti (·) is Lipschitz continuous for each i and hence so is f (·). Thus, f is also absolutely continuous and its derivative exists almost everywhere. If f˙(t) exists for t > 0 we call t a regular point. Theorem 4.1. The workload function f is a Lyapunov function for Q if ρ(M) < 1. Proof. Let H = G−1 MG. Note that ρ(H) = ρ(M) < 1. Hence, I − H is an M-matrix. Therefore I − H is invertible with a non-negative inverse. Next, recall we have W (t) = W (0) − (I − H)T (t). 3

Multiplying by e′ (I − H)−1 yields e′ (I − H)−1 W (t) = e′ (I − H)−1 W (0) − e′ T (t). Let f (t) = e′ (I − H)−1 G−1 Q(t). This function is 0 iff Q(t) = 0. Then note the following: f (t) = e′ (I − H)−1 W (t) ′

−1

= e (I − H) Taking derivatives, we have

(10) ′

W (0) − e T (t)

(11)

f˙(t) = −e′ T˙ (t) = −1,

whenever W (t) 6= 0. Therefore, f is a Lyapunov function for any feasible non-idling fluid policy. Furthermore, the draining time of the system under any policy is given by f (0) = e′ (I − H)−1 W (0) which can be interpreted as the initial unfinished “potential” work (work due to the current workload and work generated in the future by the initial workload’s “offspring”). Corollary 4.2. The fluid model is globally stable if ρ(M) < 1.

4.1

Weak Instability

Our goal is to prove that the model is weakly unstable if if ρ(M) > 1. We begin by introducing the following Lemma: Lemma 4.3. Since ρ(M) = ρ(H) > 1, by M-matrix theory, all nonnegative vectors T (t) > 0, V (t) = (I − H)T (t) must have some entry vi (t) < 0 for some i ∈ {1, ..., K}. Proof. Proof by contradiction. Assume, for sake of contradiction, that for any H s.t. ρ(H) > 1, there exists some nonnegative vector T (t) > 0, V (t) = (I − H)T (t) ≥ 0. Since each component of T (t) is strictly positive, all elements in M are assumed to all be strictly positive, which in turn implies that T˙ (t) is always strictly positive, ′ ′ ′ even when the fluid level is zero. We may then construct some H s.t hij < hij and ρ(H ) = 1. We now consider: ′ ′ ′ V (t) − V (t) = (I − H)T (t) − (I − H )T (t) = (H − H)T (t) < 0. ′

′

The above equations imply that V (t) > V (t) and that there exists some T (t) > 0 s.t (I − H )T (t) > 0. ′ ′ Thus (I − H ) is semipositive, and by condition I27 in [2], (I − H ) is a non-singular -matrix. This implies that ρ(H′ ) < 1, a contradiction. We are now ready to prove our theorem. Theorem 4.4. The model is weakly unstable if ρ(M ) > 1. Proof. We now consider Q(0) = W (0) = 0 and for any t > 0. We have that W (t) = W (0) + H(T (t)) − T (t) = (H − I)T (t). By the above Lemma, there exists some entry of W (t) s.t Wi (t) > 0. This implies that Q(t) 6= 0 for t ≥ 0. Thus the model is weakly unstable. 4

4.2

Weak Stability

Lemma 4.5. The fluid model is weakly stable if ρ(M) ≤ 1. Proof. We need to show that if Q(0) = 0, then Q(t) = 0 ∀ t > 0. Suppose Q(0) = 0 and let t be a positive real number. Note that Q(t) = (M − I)D(t)) is a non-negative matrix. As such, it suffices to show that a positive weighted sum of Q1 (t), ..., QN (t) is non-positive. Since M is a positive matrix, it follows by Perron-Frobenius that there exists a positive left eigenvector w of M with eigenvalue ρ(M) : w′ M = ρ(M)w′ , wi > 0 for 1, ..., K. Thus we have: 0 ≤ w′ Q(t) = w′ (M − I)D(t) = (ρ(M − 1))w′ D(t) ≤ 0

∀t ≥ 0.

This finishes the proof.

5

Branching Process Connections

A classical tool for the simple M/G/1 queue and related systems is to interpret customers as individuals in a branching process, such that the children of a customer is the number of customer arriving during his service. This is useful for example since the stability condition for the queuing system is the same as the condition for a.s. extinction. Carrying out the same idea for our multiclass systems leads to a (1) (K) K-type Crump-Mode-Jagers branching process {(Zn = Zn , . . . Zn ) : n ≥ 1}, such that the lifetime of an individual of type j has the same distribution as Sj . If Ei refers to a single ancestor of type i, the offspring mechanism is then described by the probabilities Z ∞
(λij s)k e−λij s (j), dFj (s) , (12) pij (k) = Pi Z1 = k = P (Pois(λij Sj ) = k) = k! 0  (j)  The offspring matrix M = (Mij )i,j=1,...,K is given by Mij = Ei Z1 = λij /µj and is assumed irreducible. Thus Perron-Frobenius theory applies to M P and nwe denote by ρ = ρ(M) the largest eigenvalue. Note that the ijthe element of the matrix ∞ of type j 0 M gives P the nexpected number −1 . progeny of an individual of type i; of course, when ρ < 1, we have ∞ M = (I − M) 0

5.1

Stability Conditions

P (j) ∗ Let |Zn | = K 1 Zn denote the total number of individuals in the nth generation and T the extinction (i) ∗ time, π = Pi (T < ∞) the extinction probabilities of the branching process. Then, by classical results, we have the following theorem: Theorem 5.1. π (i) = 1, i = 1, . . . , K, if and only if ρ ≤ 1.

(13)

Proof. See [7], Theorem 7.1. Let us consider K = 2. In this case, we can solve explicitly for ρ. By algebra, we obtain that ρ ≤ 1 is equivalent to: r 2 λ11 λ22 λ22 λ21 λ11 + + − + 4λµ12 µ1 µ2 µ1 µ2 1 µ2 ≤ 1. (14) 2 5

Theorem 5.2. Ei T ∗ < ∞ for all i f and only if ρ < 1. Proof. This is standard folklore in the branching area. For a simple proof of sufficiency, assume ρ < 1 and let Sj (m; n) denote the lifetime of the mth individual of type j in the nth generation, µ = minK 1 µj . Then (j)

Ei T

∗

≤ Ei

∞ X K Z n X X

Sj (n) = Ei

n=0 j=1 m=1

where

P∞ PK 0

1

∞ X K (j) X Zn n=0 j=1

µj

≤ µEi

∞ X K X

Zn(j) = µ

n=0 j=1

∞ X K X

Mijn < ∞

n=0 j=1

Mijn < ∞ follows from ρ < 1.

We have the following corollary to Theorem 5.1. Corollary 5.3. The busy period T < ∞ w.p.1 if and only if the matrix M given by λij Mij = , i, j = 1, . . . , K, µj has largest eigenvalue ρ(M) ≤ 1. Similarly, a corollary to Theorem 5.2 is Corollary 5.4. For the busy period T , ET < ∞ if and only if ρ < 1.

5.2

Further Applications

Let Bi;z denote the length of the busy period initiated by a class i customer with service requirement z (as before, Bi that of the standard busy period initiated by a class i customer, that is, taking z = Si ). Let further (j) ∞ Z n X X Sj (m; n) τj = E i n=0 m=1

be the expected total time in [0, Bj ) where the customer being served is of class i. As before, G is the diagonal matrix with the µi on the diagonal. Lemma 5.5. Assume ρ < a. Then: −1 −1 (i) (Ei τj )i,j=1,...,K = (I − M)−1 G−1 ; (ii) EBi = e⊤ i (I − M) G e ; ⊤ −1 −1 (iii) EBi;z = z/ρi where ρi = ei Λ(I − M) G e ; (iv) Bi;z /z → ωi in probability as z → ∞. Proof. (i) follows immediately since the ij element of (I − M)−1 G−1 is ∞ X

Mijn /µj = Ei

∞ X

Zn(j) /µj = Eτi

n=0

n=0

and (ii) follows from (i) by summing over j. For (iii) and (iv), we may by work conservation assume that the discipline is preemptive-resume. The workload process during service of a class i customer ¯ i = PN λij and c.d.f. then evolves as a standard compound Poisson process with arrival rate λ i=1 K X λij ¯ i P(Bj ≤ x) λ j=1

¯ i PK λij /λ ¯ i EBj which is the same as ρi . of the jumps. For this system, the rate of arriving work is λ 1 Now simply apppeal to standard compound Poisson results to get (iii), (iv). 6

6

Busy Period Results

First, we observe that the busy period of the system corresponding to an arbitrary initial state x = (x1 , . . . , xK ) ∈ ZK + is the sum of busy periods, each of which corresponds to the branching process starting with a single customer. Recall x = (x1 , . . . , xK ) ∈ ZK + . We provide the LST of the busy period in Theorem 6.1 below. Theorem 6.1. The LST of the busy period Bx for any nonzero x ∈ Zn+ is given by gx (s) =

N Y

zixi ,

(15)

i=1

where zi = zi (s, x) is the unique solution of the following nonlinear equation ¯j ) (s + λ

1 − ψj u−1 j 1 − ψj

=

N X

λij ui ,

0 ≤ ui ≤ 1,

j = 1, . . . , N,

(16)

i=1

¯ j ). with ψj = ψj (s + λ We introduce Lemma 6.4 below to prove this Theorem, but first make a few remarks. Remark 6.2. When the service times are exponential with rate µj , the equation (16) is simplified as N X

λij ui + µj

i=1

1 ¯j , = s + µj + λ uj

j = 1, . . . , N.

(17)

Remark 6.3. For the M/G/1 queue, when K = 1, it is well known that the LST g(s) of the busy period B is given by g(s) = ψ(s + λ − λg(s)) (18) where ψ is the LST of the service time and λ is the arrival rate. See, for example, [10] for a derivation using a branching-type argument. Our result reduces to g(s) = gx (s) = g1 (s) = z,

(19)

(s + λ)(1 − ψ(s)/z) = λz(1 − ψ(s)).

(20)

where z is a solution to We now introduce Lemma 6.4 below. Lemma 6.4. Let Bx be the LST for any nonzero x ∈ ZK + . Let ei be the N -vector with all components zero except the ith component equal to one. For any nonzero x ∈ ZK + , if a job of class j is in service, j j the Laplace transforms gx of Bx satisfies ¯j ) + gx (s) = gx−ej (s)ψj (s + λ

N X

gx+ei (s)

i=1

7


λij ¯j ) , 1 − ψ (s + λ j ¯j s+λ

s > 0.

(21)

Proof of Lemma 6.4. Suppose the system at time t = 0 is in state x and is starting to serve a job of class j. Note that the busy period remains the same, regardless which class of job to start serving first. Let us denote τia to be the arrival time for the first job of class i, i = 1, ..., N and τ d to be the a , τ d ), and let τ 2 be the remaining departure time of the first job in service. Define τ 1 = min(τ1a , ...τN time to empty after the first arrival or departure event. Note that τ 1 and τ 2 are independent. Then we can write Bx = τ 1 + τ 2 . By conditioning, we obtain the following Z ∞   e−st P τ d ∈ dt, τia > t, i = 1, ..., N gx (s) = gx−ej (s) 0

+

N X

gx+ei (s)

i=1

= gx−ej (s) +

N X

Z

∞

∞ 0

  e−st P τ d > t, τka > t, k 6= i, τia ∈ dt

¯j

e−st e−λ t dFj (t)

0

gx+ei (s)

i=1

Z

Z

∞ 0

¯j

e−st Fjc (t)λij e−λ t dt.

¯ j ). For the second term, by integration by parts, we easily The first term is equal to gx−ej (s)ψj (s + λ obtain the second term on the right hand side of (21). This completes the proof. With the above lemma established, we are now ready to prove Theorem 6.1. Proof. By plugging (15) into (21), we obtain N Y

zixi =

N Y

¯j ) zixi zj−1 ψj (s + λ

i=1

i=1

+

N N Y X

zkxk zi


λij ¯j ) , 1 − ψ (s + λ j ¯j s+λ

N X


λij ¯j ) . 1 − ψ (s + λ j ¯j s+λ

i=1 k=1

which implies that 1=

zj−1 ψj (s

¯j ) + +λ

zi

i=1

By simple algebra, we obtain (16).

7

Busy Period Asymptotics

We present here some preliminary results on the tail asymptotics of the busy period in the case of heavy-tailed service time distributions. For recent work on light tails and references, see Palmowski and Rolski [11]). For the present case of heavy tails, particular relevant references are Zwart [12], Jelenkovi´c and Momcilovi´c [8] and Denisov and Shner [4]. 8

Whereas [4] builds on particular structure of the M/G/1 queue not easy to generalize to the present model, the key idea in [12], [8] is (as in many other instances of heavy-tailed behaviour) the principle of one big jump. For busy periods, this leads to expect a large busy period to occur as consequence of one big service time. Consider for simplicity the standard M/G/1 queue and say there is a single large service time, say of size S = z. The workload right after the big jump is u + z for some small or moderate u and then decreases at rate 1 − ρ until 0 is hit and the busy period terminates. Appealing to the LLN, the time for this is appr. (z + u)/(1 − ρ), and since the time before the big jump can be neglected, we have B > x iff z > (1 − ρ)x. Now note that (Asmussen [1], Foss and Zachary [5]) the probability of such a big jump is Eσ where σ is the expected number of customers served in a busy period. Inserting Eσ = 1/(1 − ρ) leads to
1 · P S > (1 − ρ)x 1−ρ

P(B > x) ∼

(22)

which can indeed be shown to be the correct asymptotics if the √ service time distribution is subexponential and square root insensitive, i.e. with heavier tail than e− x ([8]). P K Generalizing this approach to our multiclass system, recall that ρi = 1 λij EBj and introduce a subexponential and square root insensitive reference distribution F to which the individual service time distributions are related by means of
F i (1 − ρi )x ∼ ci F (x) (23)
(in practice, one chooses F as supi F i (1 − ρi )x ). This is a common device in heavy-tailed studies involving distributions with different degrees of heavy-tailedness and in particular, it allows some Fj to be light-tailed (then cj = 0). Recalling the interpretation of ρi as the rate of arriving work while a class i customer is in service, a big service time Sj of a class j customer will lead to Bj > x precisely when Sj > 1 − ρi . The probability that this occurs for one of the service times in [0, Bi ) is approximately ∞ X K X

mnij F i (1 − ρi )x

n=0 j=1




∼

∞ X K X

mnij cj F (x) = di F (x) (say).

n=0 j=1

As for the standard M/G/1 queue, it is in fact a matter of routine to verify that this is an asymptotic lower bound: Proposition 7.1. Assume that ck > 0 for some k. Then for each i = 1, . . . , K, lim inf x→∞

P(Bi > x) ≥ di . F (x)

(24)

Further, the M/G/1 literature leads to the conjecture that this is also an asymptotic upper bound, i.e. that P(Bi > x) ∼ di F (x). However, the upper bound is more difficult already for M/G/1 and is currently being detailed in the multiclass setting by the authors. Remark 7.2. The assumption F i (x) ∼ e ci F 0 (x)

(25)

may apriori be more appealing than (23) since it does not evaluation of the ρi . But it is of course closely related: if F is regularly varying with F (x) = L(x)/xα , then (23) and (25) with F0 = F are 9

equivalent, with the constants connected by cj = e ci /(1 − ρj )α . For F0 lognormal or Weibull with tail
β e−x (where β < 1/2 in the square root insensitive case), one has F 0 (γ1 x) = o F 0 (γ 2 x) when γ1 > γ2 .
∗ ∗ Hence if (25) holds, we may define ρ∗ = maxK 1 ρj and take F (x) = F 0 (1 − ρ )x , cj = 1 if ρj = ρ , cj = 0 if ρj < ρ∗ .

8

Acknowledgments

We thank Professor Gordan Pang and Yizhou Xia for helpful conversations.

References [1] S. Asmussen (1998) Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. The Annals of Applied Probability 8, 354– 374 [2] A. Berman and R.J. Plemmons (1979) Nonnegative Matrices in the Mathematical Sciences. Academic Press. [3] M. Bramson (2008) Stability of Queueing Networks. Springer. [4] D. Denisov & S. Shneer (2010) Global and local asymptotics for the busy period of an M/G/1 queue. Queueing Systems 64: 383–393. [5] S. Foss and S. Zachary (2003) The maximum on a random time interval of a random walk with long-tailed increments and negative drift The Annals of Applied Probability 13, 37–53. [6] D. Gamarnik (2010) Fluid models of queueing networks. Wiley Encyclopedia of Operations Research and Management Science. [7] T.E. Harris (1963) The Theory of Branching Processes. Springer-Verlag. [8] P. Jelenkovi´c and P. Momcilovi´c (2004) Large deviations of square root insensitive random sums. Mathematics of Operations Research 29: 398–408. [9] D.R. Miller (1981) Computation of steady-state probabilities for M/M/1 priority queues. Operations Research 29: 945-958. [10] M. F. Neuts (1974) The Markov renewal branching process. In Proc. Conf. Mathematical Methods in the Theory of Queues, Kalamazoo. [11] Z. Palmowski and T. Rolski (2006) On the exact asymptotics of the busy period in GI/G/1 queues. Advances in Applied Probability 26: 485–493. [12] B. Zwart (2001) Tail asymptotics for the busy period in the GI/G/1 queue. Mathematics of Operations Research 38: 792–803.

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