Sojourn times in finite-capacity Processor-Sharing queues - CiteSeerX

Sojourn times in finite-capacity Processor-Sharing queues Sem Borst∗§† ∗ CWI

P.O. Box 94079 1090 GB Amsterdam The Netherlands [email protected] § Bell Labs, Lucent Technologies

Onno Boxma∗†‡

† Department

of Mathematics and Computer Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands [email protected]

Abstract— Motivated by the need to develop simple parsimonious models for evaluating the performance of wireless data systems, we consider finite-capacity ProcessorSharing systems. For such systems, we analyze the sojourn time distribution, which presents a useful measure for the transfer delay of documents such as Web pages. The service rates are allowed to be state-dependent, to capture the fact that the throughput in wireless data systems may vary with the number of active users due to scheduling gains or channel collisions. We derive a set of linear equations for the Laplace-Stieltjes transforms of the sojourn time distributions, conditioned on the number of users upon arrival. This set of equations is solved, and the resulting LST’s are inverted, resulting in a phase-type distribution for the unconditional sojourn time. Numerical results are provided, and two types of approximations are proposed that substantially reduce the computational effort.

Keywords: wireless data systems, Processor Sharing, sojourn time distribution I. I NTRODUCTION In the present paper we analyze sojourn time distributions in finite-capacity Processor-Sharing (PS) systems. The primary motivation for our study arises from the need to develop simple parsimonious models for evaluating the performance of wireless data systems while capturing the vital characteristics of wireless communications. A particularly crucial feature of wireless networks is the fact that channel conditions show huge random variations across both space and time. Wireless circuit-switched voice systems rely on power control algorithms for adjusting the transmit power so as to compensate for the varying channel quality and maintain

Nidhi Hegde‡

‡ EURANDOM

P.O. Box 513 5600 MB Eindhoven The Netherlands [email protected]

a constant transmission rate. Various data applications on the other hand, such as file transfers and Web browsing sessions, do not have a stringent rate requirement and are less sensitive to packet-level delays. Such elastic applications are well-suited for rate control algorithms which adapt the transmission rate over time to track the fluctuations in the channel quality while transmitting at constant (maximum) power. In the latter scenario, the varying channel conditions manifest themselves in fluctuations in the feasible transmission rates. This has triggered a vast interest in so-called channel-aware or opportunistic scheduling algorithms, which aim to improve the throughput performance by scheduling the data transmissions to the various users when their rates are relatively favorable [1]–[6]. A particularly popular algorithm is the Proportional Fair scheduling discipline, which is the default scheduler implemented in the CDMA 1xEV-DO system [7]–[10] and is also considered for implementation in HSDPA. It has been shown in [11] that the flow-level performance of the Proportional Fair algorithm may be evaluated by means of a PS model where the total service rate varies with the total number of users. The statedependent service rates account for the scheduling gains which increase with the number of active users: the larger the number of users, the greater the opportunity for throughput gains from selecting a user with a high transmission rate. The PS model provides simple, explicit expressions for key performance measures such as the distribution of the number of active users, the mean sojourn time, and the user throughput. In particular, these performance measures are insensitive, in the sense that

they only depend on the service requirement distribution through its mean. A similar PS model for evaluating the flow-level performance of wireless LAN’s has been presented in [12]. It is worth observing that in the latter model the service rates decrease with the number of active users, reflecting the throughput losses due to the increase in channel collisions. Several related PS models for various types of wireless data systems have been proposed in [13]–[18]. It is worth emphasizing that the PS model applies even when the users have totally different transmission rates, as is typically the case for spatially diverse users, as long as all users receive an equal fraction of the transmission resources. The latter property obviously holds for a channel-oblivious round-robin scheduling strategy. Interestingly enough, it also holds for the above-mentioned Proportional Fair algorithm, provided the relative rate fluctuations around the long-term average values are statistically identical, see for instance [9], [11]. Even when the latter mild assumption is only approximately satisfied, the PS model usually continues to provide a reasonable approximation. In the present paper we examine sojourn time distributions in finite-capacity PS systems with state-dependent service rates. The state-dependent service rates serve to capture the fact that the throughput in wireless data systems may vary with the number of active users due to scheduling gains or channel collisions as described above. Our model also incorporates a finite capacity constraint in order to model a further important feature of wireless systems: because of a limited number of available code words and feedback links, only a certain maximum number of users can be simultaneously supported, even in the absence of any explicit admission control. In contrast to the queue length distribution and the mean sojourn time, the sojourn time distribution in a PS system does not have a simple form, and is quite sensitive to the service requirement distribution. In fact, the sojourn time distribution has remained largely intractable, even in the case of a fixed service rate and infinite capacity. For the M/M/1 PS queue, Coffman et al. [19] derived a closed-form expression for the Laplace-Stieltjes Transform (LST) of the sojourn time distribution conditioned on the service requirement and the number of customers seen upon arrival. Sengupta & Jagerman [20] found an alternative expression for the LST of the distribution of the sojourn time conditioned only on the number of customers seen upon arrival. Building on [19], Morrison [21] obtained a compli-

cated integral expression for the distribution function of the sojourn time. For results on sojourn time distributions in M/G/1 PS queues, we refer to the survey papers [22], [23]; see Ramaswami [24] and Jagerman & Sengupta [25] for the G/M/1 PS queue. In the present paper we derive a set of linear equations for the LST’s of the conditional sojourn time distributions (conditioned on the number of users upon arrival). These equations can be extended from exponential service requirements to Coxian distributions by conditioning on the relevant service phases as well. Solving the set of linear equations, the LST’s can be expressed in terms of the roots of the determinants of the involved matrices. By inverting the LST’s and numerically computing the roots, the sojourn time distribution can then be calculated to any degree of accuracy. The remainder of the paper is organized as follows. In Section II we present a detailed model description and focus on the case of exponentially distributed service requirements. We consider the case of a Coxian distribution in Section III. In Section IV we provide some numerical results and discuss two types of approximations that substantially reduce the computational effort involved in finding the appropriate roots. II. T HE M/M/1/N PS

QUEUE

We first consider the M/M/1/N PS queue. In this system, customers arrive according to a Poisson process with rate λ. Customers in the system are served in a Processor-Sharing manner, with total service rate fi when there are i customers. Such a service discipline is called Generalized Processor Sharing by Cohen [26], and the choice fi = 1 reduces the service discipline to ordinary Processor Sharing. When there are N customers in the system, any new arrivals are blocked and cleared. The service requirements follow an exponential distribution with mean 1/µ. We will consider more general service requirement distributions in the next section. Let Si denote the sojourn time of a customer given that it finds i − 1 other customers in the system upon arrival. We define ψi (ω) = E[e−ωSi ] to be the LST of the sojourn time distribution conditioned on an arrival finding i − 1 other customers. Once we obtain the distribution of Si , we can easily derive the unconditional sojourn time distribution, using the distribution of the number of customers in the system at arrival epochs. Since the arrivals follow a Poisson process, the steady-state distribution of the number of customers at arrival epochs is equal to that at arbitrary instants. This probability distribution has been derived in [26]. Denoting Xarr to

be the number of customers in the system seen by an arrival, the unconditional sojourn time distribution is as follows: P[S > t] =

N −1 X

P[Xarr = i]P[Si+1 > t]

i=0

= H −1

N −1 X

ρi φ(i)P[Si+1 > t],

i=0

´−1

i where ρ = λ/µ, φ(i) = , and H = k=1 fk PN −1 j j=0 ρ φ(j). Note that H is a normalizing factor so that the sojourn time is conditioned on the customer having entered the system, thus not blocked. Using the memoryless property of the exponential distributions for interarrival times and service requirements, we have the following recurrence relations for ψi (·):

³Q

λ f1 µ ψ2 (ω) + , h1 (ω) h1 (ω) i − 1 fi µ λ ψi+1 (ω) + ψi−1 (ω) ψi (ω) = hi (ω) i hi (ω) 1 fi µ + , i = 2, . . . , N − 1, i hi (ω) λ N − 1 fN µ ψN (ω) + ψN −1 (ω) ψN (ω) = hN (ω) N hN (ω) 1 fN µ + , (1) N hN (ω) ψ1 (ω) =

where hi (ω) = λ + fi µ + ω. Indeed, consider for example the relation for ψi (ω), i = 2, . . . , N − 1, for some tagged customer that sees i − 1 other customers upon arrival. Since the processor is shared equally by all customers, the sojourn time of this test customer depends on the presence of other customers, and is thus affected by arrivals and departures. Since interarrival times and service requirements are exponentially distributed, the time to the next event is exponential with rate λ + fi µ for i < N , and with rate fi µ for i = N . If the next event is an arrival, then the sojourn time of the test customer includes the time to this event plus the sojourn time as if it had arrived to a system with this additional customer. This is represented by the first term on the right-hand side in the second relation (1). Similarly, if the next event is a service completion of some other customer, then the sojourn time includes the time to this event plus the sojourn time as if the test customer had arrived to meet i − 1 customers, as shown by the second term. The last term represents the next event being the departure of the test customer. Note that these relations are identical to those of waiting times in

a queue with a random order of service (ROS) discipline with service speed fi when there are i customers waiting, i.e., if a departure occurs all customers present have an equal chance of being the one to depart. Remark 1: It has been shown [27] that for the G/M/1 queue, the sojourn time under the PS discipline is equal in distribution to the waiting time in a queue under the ROS discipline, conditioned on this waiting time being positive, i.e., P[S > t] = P[WROS > t|WROS > 0]. In our model, the service speed fi destroys this exact translation, so that the speed fi corresponds to i customers in the system with the PS discipline and i customers waiting in the queue with the ROS discipline. Furthermore, in order to uncondition, P[Si+1 > t] is multiplied by P[Xarr = i] under the PS discipline and the conditional waiting-time distribution is multiplied by the probability that the arriving customer meets i waiting customers under the ROS discipline. The relations in (1) form a system of N linear equations given by M Ψ(ω) = R, where 

M

   =  

λ + f1 µ + ω .. . i−1 − i fi µ .. .

−λ .. . λ + fi µ + ω .. .



..

. −λ .. . N −1 − N fN µ

fN µ + ω

  ,   

Ψ(ω) = (ψ1 (ω), . . . , ψN (ω))T , fN µ T fi µ ,..., ) . i N The matrix M is strictly diagonally dominant, that is, P |mii | > j,j6=i |mij | where mij is the (i, j)th element of M . M is thus non-singular and det M 6= 0 [28]. We can then use Cramer’s rule to get the following solution [28]: det Mi ψi (ω) = , (2) det M where the matrix Mi is equal to M with the ith column replaced by the vector R. It can easily be seen that the denominator of ψi (ω) is a polynomial of degree N in ω . That is, det M = ω N + · · · + c1 ω + c0 , where ci are corresponding coefficients. We can then write the solution to ψi (ω) as follows: and

R = (f1 µ, . . . ,

ψi (ω) =

N X

Ai,j , ω − ωj j=1

i = 1, . . . , N,

(3)

where ωj , j = 1, . . . , N , are the N roots of det M and Ai,j are the residues of the partial-fraction expansion of (2) for the given roots. Let C = ωI − M , where I is

an identity matrix of appropriate size. Using Gerˇsgorin’s Theorem [28], it can be seen that the eigenvalues of C , and thus the roots of det M , lie in the left half plane. Furthermore, it can be shown that these roots are real and unique. Consider the matrix C and denote ci,j to be the element in the (i, j)th position. Since C is a real and tridiagonal matrix, and ci,i−1 ci−1,i > 0, i = 2, . . . , N, all eigenvalues of C are real and unique [29]. Specifically, it can be shown using Gerˇsgorin’s Theorem −3 N −1 that −2λ + min[− 2N N −1 fN −1 µ, − N fN µ − fN −1 µ] ≤ Nµ . wj ≤ − fN Now, inverting the transform (3), we have1 : P[Si > t] =

N X

Ai,j ωj t e . −ω j j=1

(4)

Remark 2: A similar result for the waiting-time distribution in an M/M/1/N system with a ROS discipline has been derived in [30]. Clearly, P[Si > t] is a phase-type distribution. With hindsight, this is not surprising, as the following reasoning shows. It is known that a phase-type distribution represents the distribution of the time to absorption of a Markov process [31]. Consider a Markov process with states 1, . . . , N , corresponding to the tagged customer being one out of 1, . . . , N customers, and with a state 0, an absorbing state where the tagged customer leaves the system after having completed service. When there is an arrival to the system, this process moves from some state i to i + 1, i < N ; when there is a departure of one of the other customers, the process moves from state i to i−1, i > 0; and finally as the tagged customer leaves the system, the process jumps into the absorbing state. The sojourn time of a customer can then be viewed as the time to absorption in a Markov process. We demonstrate this using a simple example of a system with a maximum of N = 2 customers. Example (N = 2): From (2), we have the following: ψ1 (ω) = ψ2 (ω) =

f1 µ(f2 µ + ω) + 21 λf2 µ , (λ + f1 µ + ω)(f2 µ + ω) − 12 λf2 µ 1 2 f2 µ(λ

+ f1 µ + ω) + 12 f1 µf2 µ . (λ + f1 µ + ω)(f2 µ + ω) − 12 λf2 µ

Now let Bik denote the time that the tagged customer spends in the system with i−1 other customers, state i in 1 roots the sojourn time density equals PnIf the Pnj are not tunique, k−1 ωj t

e A , where n is the number of distinct j=1 k=1 i,j,k (k−1)! roots, and nj is the multiplicity of root j.

the Markov process described above, during its k th pass through this state. Since N = 2, the tagged customer switches between states 1 and 2 until it completes service. Its sojourn time in the system is then the sum of the times spent in these states, that is, S1 = B11 + B21 + · · · + B1J , or S1 = B11 + B21 + · · · + B1J + B2J when the tagged customer makes J passes through state 1. Denote ψ˜i (ω) to be the LST of the time to absorption given that the process starts in state i. For state 1, this transform is as follows: ψ˜1 (ω) = +

∞ X

k

(q1 q2 ) q¯1

k=0 ∞ X

µ

λ + f1 µ λ + f1 µ + ω

(q1 q2 )k q1 q¯2

k=0

µ

¶k+1 µ

f2 µ f2 µ + ω

f2 µ λ + f1 µ λ + f 1 µ + ω f2 µ + ω

¶k

¶k+1

,

where qi is the probability that the process jumps from state i to some other state, and q¯i is the probability that the process jumps from state i to the absorbing state, indicating the tagged customer has completed service. f1 µ λ Specifically, q1 = λ+f , q¯1 = λ+f , and q2 = q¯2 = 1µ 1µ 1/2. With some algebraic manipulation, we find that ψ˜i (ω) = ψi (ω). The observation that the sojourn time may be interpreted as the time to absorption in a Markov process in fact provides an alternative derivation of (1). It may easily be checked that (C, R) is the generator matrix of the above-mentioned Markov process, with the last column corresponding to the absorbing state. Equation (1) agrees with the expression for the LST of the absorption time distribution in terms of (C, R) provided in [32]. III. C OXIAN

SERVICE REQUIREMENT DISTRIBUTION

We now consider more general service requirement distributions. Specifically, the service requirement distribution is assumed to be Coxian with K phases, as shown in Figure 1, where 1/µk is the mean service requirement in phase k , pk is the probability of going to phase k + 1 after completion in phase k , and p¯k is the probability of completing total service after service in phase k . Denote Xk,j to be the number of customers in phase j just prior to the arrival of a test customer into phase k , ~ k = (Xk,1 , . . . , Xk,K ). Let Sk,~ı be the remaining with X sojourn time of a customer given that it arrives to phase k with (Xk,1 = i1 , . . . , Xk,k = ik − 1, . . . , Xk,K = iK ) customers in the system, and let ψk,~ı(·) be the LST of the corresponding distribution. For service requirement distributions that follow a K stage Coxian distribution, the recurrence relations for

1 µ1

2 µ2

p1

p2

...

pK-1

p2

p1

Fig. 1.

K µK

f|~ı|

pK-1

pK

Coxian Distribution

ψk,~ı(·) are as follows: (λ + g(~ı, µ ~ ) + ω)ψk,~ı(ω) = λψk,~ı+e1 (ω) K X µ f|~ı| ij µj ¶ ¡ pj ψk,~ı−ej +ej+1 (ω) + |~ı| j=1,j6=k

¢ + (1 − pj )ψk,~ı−ej (ω) µ ¶ ¶µ f|~ı| ik µk ik − 1 ¡ + pk ψk,~ı−ek +ek+1 (ω) |~ı| ik + (1 − pk )ψk,~ı−ek (ω)) ¶µ ¶ µ ¢ ¡ f|~ı| ik µk 1 + pk ψk+1,~ı−ek +ek+1 (ω) + 1 − pk , |~ı| ik |~ı| = 1, . . . , N − 1, (g(~ı, µ ~ ) + ω)ψk,~ı(ω) = ¶ µ K X f|~ı| ij µj ¡ pj ψk,~ı−ej +ej+1 (ω) |~ı| j=1,j6=k

¢ + (1 − pj )ψk,~ı−ej (ω) µ ¶ ¶µ f|~ı| ik µk ik − 1 ¡ + pk ψk,~ı−ek +ek+1 (ω) |~ı| ik + (1 − pk )ψk,~ı−ek (ω)) µ ¶µ ¶ ¢ f|~ı| ik µk 1 ¡ + pk ψk+1,~ı−ek +ek+1 (ω) + 1 − pk , |~ı| ik |~ı| = N, (5)

k = 1, . . . , K − 1,

(λ + g(~ı, µ ~ ) + ω)ψK,~ı(ω) = λψK,~ı+e1 (ω) K−1 X µ f|~ı| ij µj ¶ ¡ + pj ψK,~ı−ej +ej+1 (ω) |~ı| j=1 f|~ı| iK µK + |~ı| |~ı| = 1, . . . , N − 1, µ

¶µ

PK

i j µj

j=1 where g(~ı, µ ~) = , µ ~ = (µ1 , . . . , µK ), ek |~ı| is a vector of zeros with a one at position k , and P |~ı| = K j=1 ij . We thus have a set of recurrence relations, where the relations for phase k , ψk,~ı(ω), ∀~ı : |~ı| ≤ N depend on those for phase k +1, ψk+1,~ı(ω), ∀~ı : |~ı| ≤ N . The equations for the last phase, K , can be solved independently of the other phases. As shown in (2) in Section II, we can solve for ψK,~ı in a similar manner, using the determinant of the coefficient matrix for the relations (6). The denominator of ψK,~ı(ω) is then a ¡ ¢ polynomial of degree LK = N +K−1 . The solution K for ψK,~ı(ω) can thus be written in a form similar to P K A(K,~ı),j (3), ψK,~ı(ω) = L j=1 ω−ωK,j , |~ı| ≤ N , where ωK,j are the zeros of the determinant for the coefficient matrix for the system of equations (6) for phase K . This can now be substituted into the equations for phase K − 1. The resulting set of linear equations, which include the solution to ψK,~ı, can be solved in an identical manner as shown for phase K . This process is continued for all other phases, k = K − 2, . . . , 1, to finally arrive at the solution for ψ1,~ı. Each set of linear equations in this process is of size LK . We have thus derived the sojourn time distribution conditioned on the number of customers in the system just before an arrival. The distribution of the unconditioned total sojourn time in the system is then as follows:

P[S > t] =

X

~ 1 = ~ı − e1 ]P[S1,~ı > t]. P[X

|~ı| t] φ(|~ı| − 1)

|~ı| t] against t for exponential service requirements. Figure 2 shows the complementary sojourn time distribution P[S > t] for N = 20, for various fi . We assume a load of ρ = 0.8, with 1/µ = 1. The case fi = 1 corresponds to the egalitarian PS discipline. The case fi = log i+EG corresponds to a particular weight-based scheduling strategy on the downlink of a wireless data system as considered in [11]. For this wireless system, define Rk to be the feasible rate of a user k in a given time slot. The feasible rate of a user fluctuates over time due to signal fading. Define Ck to be the time-average rate of user k . Assume that the relative fluctuations of the feasible rates for the various users around their respective time-average rates are independent and identically d distributed. We can then write Rk = Ck Yk , where Yk corresponds to the variation in the feasible rate, and Y1 , . . . , Yi , when there are i users in total, are independent and identically distributed copies. The weight-based scheduling strategy considered in [11] assigns a given Rj k time slot to user k when R Ck = maxj=1,...,i Cj , when there are i users in total. This amounts to selecting user k when Yk = maxj=1,...,i Yj . The (long-term) expected rate received by user k then is Ck fi /i, where fi = E[maxj=1,...,i Yj ] represents the relative throughput gain due to this type of channel-aware scheduling. When Yk is assumed to follow Rayleigh fading, this gain factor is fi = 1 + 1/2 + . . . + 1/i, which may be approximated by fi ≈ log i + γ for large values of i, where γ = 0.57721 is Euler’s constant, and is denoted by EG in Figure 2. P The figure also plots the distribution for fi = ik=1 1/k , referred to as sum(1/k,i) in the figure. The approximation P of fi = ik=1 1/k ≈ log i + γ appears to be quite close. Figure 3 compares the sojourn time distribution for N = 20 with that of an infinite-capacity PS queue derived by Morrison [21]. The sojourn time distribution obtained in [21] is quite close to that of the finite case,

Fig. 2.

10

15

20

Complementary Sojourn Time Distribution, N=20

1

N=20

0.8 N=inf 0.6 0.4 0.2 0

0

Fig. 3.

5

10

15

20

Complementary Sojourn Time Distribution

suggesting that even for moderately large N , the sojourn time distribution can be approximated reasonably well by the closed-form expression for the infinite-capacity PS queue. A. Approximations Calculating the sojourn time distribution can become quite computationally expensive as N increases. The most demanding computations involve finding the N roots ωj , j = 1, . . . , N and the N residues Ai,j , i = 1, . . . , N, corresponding to each of these roots. In this section we consider two types of approximations, based on minimizing one or both of these steps. The first approximation, denoted by A, uses only the smallest (in absolute value) root. Denote this root by ω1 . The sojourn time distribution is thus approximated as follows: PA [S > t] = eω1 t . (7) Since this root corresponds to the smallest rate in the sum of exponentials shown in (4), we expect that the approximation is more accurate as t → ∞.

1

1 Exact

0.8

Exact

0.8

App B1 0.6

App B1 0.6

App B2 App A

0.4

App B2 App A

0.4

0.2

0.2

0

Fig. 4.

5

10

15

20

25

30

0

Approximations for fi = 1, N = 20

Fig. 5.

The second type of approximation is based on the location of the roots. We observe that the intervals between neighboring roots ωj and ωj+1 , j = 1, . . . , N − 1, seem to be almost equal in length – but different from the interval between zero and the root closest to zero, ω1 . The length of the interval between the approximated roots may then be approximated using ω1 and some other root and its index. We thus have a set of approximations of this type, denoted by Bm , m = 1, . . . , N − 1. Approximation Bm corresponds to the following approximation of the roots: ω ˜ j,(m) = ω1 + (j − 1)∆m , where |ωm+1 −ω1 | . The residues corresponding to these ∆m = m approximated roots, A˜i,j,(m) , i = 1, . . . , N for root ω ˜ j,(m) , will still need to be computed, including further, N the normalization factor Dm,i = j=1 1, . . . , N , so that we have the following:

P

−1 PBm [Si > t] = Dm,i

N ˜ X Ai,j,(m)

−˜ ωj,(m) j=1

˜i,j,(m) A −˜ ωj,(m) ,

i =

eω˜ j,(m) t .

Figure 4 compares these approximations for fi = 1. For approximations of type Bm , the larger the number of roots considered, the closer are the approximations. Figure 5 shows that the approximations are even better for fi = log i + γ . V. C ONCLUSION We have derived the sojourn time distribution in finitecapacity PS systems, with state-dependent service rates. The sojourn time distribution has been represented as a phase-type distribution, whose parameters correspond to eigenvalues of certain matrices representing the system. We have proposed efficient approximations that reduce the computational complexity. In future work we propose extensions to networks of finite-capacity PS nodes and general renewal arrival processes.

5

10

15

20

Approximations for fi = log i + γ, N = 20

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