Queueing Syst (2006) 52: 281–286

DOI 10.1007/s11134-006-6264-1

A sample path relation for the sojourn times in G/G/1 − P Ssystems and its applicationsAndreas Brandt · Manfred Brandt

Received: 27 June 2003 / Revised: 18 October 2005C© Springer Science + Business Media, LLC 2006

Abstract For the single server system under processor shar-

ing (PS) a sample path result for the sojourn times in a busy

period is proved, which yields a sample path relation between

the sojourn times under PS and FCFS discipline. This relation

provides a corresponding one between the mean stationary

sojourn times in G/G/1 under PS and FCFS. In particular,

the mean stationary sojourn time in G/D/1 under PS is given

in terms of the mean stationary sojourn time under FCFS,

generalizing known results for G I/M/1 and M/G I/1. Ex-

tensions of these results suggest an approximation of the

mean stationary sojourn time in G/G I/1 under PS in terms

of the mean stationary sojourn time under FCFS.

Keywords G/G/1 . G/D/1 . G/G I/1 . Processor

sharing . Sojourn time . Waiting time . Busy period .

Sample path . First come first served . Comparative

analysis . Mean sojourn time approximation

Mathematics Subject Classification (MSC 2000)60K25· 68M20· 60G17· 60G10

1. Introduction

In this paper we consider a single server system where the re-

quests are served under processor sharing (PS) discipline, i.e.,

This work was supported by a grant from the Siemens AG.

A. Brandt (�)Wirtschaftswissenschaftliche Fakultat, Humboldt-Universitat zuBerlin, Spandauer Str. 1, D-10178 Berlin, Germanye-mail: brandt@wiwi.hu-berlin.de

M. BrandtKonrad-Zuse-Zentrum fur Informationstechnik Berlin (ZIB),Takustr. 7, D-14195 Berlin, Germany

if n (>0) requests are in the single server then each request

receives 1/n of the service capacity.1 Besides the PS disci-

pline, we consider the corresponding single server system

with infinite waiting room under the first come first served

(FCFS) discipline.

Since the PS and FCFS discipline are work conserving dis-

ciplines, their corresponding work load processes and hence

busy periods are identical. However, this is clearly not the

case for the sojourn times of requests, which are very sensi-

tive with respect to the service discipline.

Processor sharing systems with stochastic arrival and ser-

vice processes have been studied extensively in many papers,

see e.g. [18] and the references therein. Some recent papers

are [4, 5, 12, 16, 19]. Since the purpose of these studies mainly

was to determine sojourn time characteristics of a request,

e.g. its mean and variance (given its service requirement),

independence as well as distributional assumption are sup-

posed. However, in the general case, few structural properties

seem to be known. For an analysis of the general processor

sharing system there seems to be no other tool than sample

path analysis. For corresponding results see [2, 6].

The aim of this paper is to prove in Section 2 a sample path

result (Lemma 2.1) for the sojourn times of a busy period in

a single server system under PS, and as an application there

is given a relation (Theorem 2.1) between the sojourn times

under PS and FCFS discipline. In Section 3 we consider the

stable G/G/1 system under PS and FCFS. From Theorem 2.1

we obtain a relation between the mean stationary sojourn

times in G/G/1 under PS and FCFS (Theorem 3.1), and in

particular a formula for the mean stationary sojourn time in

G/D/1 under PS in terms of the mean stationary sojourn time

under FCFS, generalizing corresponding known results for

1 This type of PS discipline is often called the egalitarian processorsharing discipline, cf. [18] p. 102.

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282 Queueing Syst (2006) 52: 281–286

G I/M/1 and M/G I/1. Then, extensions of the formula are

given for a large subset of G/G I/1 systems (Theorem 3.2),

suggesting an approximation of the mean stationary sojourn

time in G/G I/1 under PS in terms of the mean stationary

sojourn time under FCFS (Remark 3.2).

2. Sample path relations for the sojourn times in abusy period under PS and FCFS

For the single server system under PS discipline let us con-

sider a sample path of a finite busy period2 during which

n requests arrive and are served. Denote by vk and sk ,

k = 1, . . . , n, the sojourn time and service requirement of

the k-th arriving request ordered according to their arrivals,

respectively. Note that zero service requirements and equal

arrival instants are allowed. Further, let v∗k the sojourn time

of the k-th request if the arrival process is stopped after the

arrival of the k-th request.

Lemma 2.1. It holds

n∑k=1

(vk − sk) = 2n∑

k=1

(v∗k − sk). (2.1)

Proof: The proof will be given by induction on n. Since for

n = 1 it holds v1 = s1 = v∗1 , obviously (2.1) is valid. Assume

now that (2.1) is true for busy periods for which at most

n requests arrive. Consider an arbitrary busy period during

which n + 1 requests are served and which arrive at the time

instants

0 = t1 ≤ t2 ≤ · · · ≤ tn ≤ tn+1

and have service times sk , k = 1, . . . , n + 1. The departure

instant τk of the k-th request is given by

τk = tk + vk, k = 1, . . . , n+1, (2.2)

cf. Fig. 1. Note that the busy period has length∑n+1

k=1 sk in

view of the work conservation law. The induction step is

divided into four steps.

1. Consider the modification of the busy period where the

(n + 1)–st arriving request is not considered. Hence the mod-

ified busy period consists of n requests. Denote the corre-

sponding sojourn times of the modified busy period by vk ,

k = 1, . . . , n. Then the modified departure times τk are given

by

τk = tk + vk, k = 1, . . . , n. (2.3)

2 See e.g. [9] p. 11.

Fig. 1 Sample path of a busy period with four requests

Applying the induction assumption to the modified busy pe-

riod and taking into account that v∗k , k = 1, . . . , n, does not

depend on the requests arriving after the k-th request, we

obtain

n+1∑k=1

(vk −sk)

=n∑

k=1

(vk − sk) +n∑

k=1

(vk −vk) + (vn+1 − sn+1)

= 2n∑

k=1

(v∗k − sk) +

n∑k=1

(vk − vk) + (vn+1 − sn+1).

(2.4)

2. Let π be a permutation of {1, . . . , n} such that

τπ (1) ≥ τπ (2) ≥ · · · ≥ τπ (n).

In the following we will exploit several times the fact that

under the PS discipline the service capacity is divided equally

among the requests being in the system, which implies that

the residual service times of the requests in the system are

reduced with equal speed. Therefore it holds

τπ (1) ≥ τπ (2) ≥ · · · ≥ τπ (n), (2.5)

too. In view of (2.5), there are indices i, j ∈ {0, . . . , n} such

that

τπ (1) ≥ · · · ≥ τπ ( j) ≥ tn+1 > τπ ( j+1) ≥ · · · ≥ τπ (n),

τπ (1) ≥ · · · ≥ τπ (i) ≥ τn+1 > τπ (i+1) ≥ · · · ≥ τπ (n),

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Queueing Syst (2006) 52: 281–286 283

respectively. Obviously, 1 ≤ j ≤ n and 0 ≤ i ≤ j . Further,

it holds

τπ (k) = τπ (k), k > j, (2.6)

τπ (k) − τπ (k+1) = τπ (k) − τπ (k+1), k < i, (2.7)

in view of the PS discipline.

For the second summand on the r.h.s. of (2.4) from (2.2),

(2.3), (2.6) and (2.7) we obtain

n∑k=1

(vk − vk)

=n∑

k=1

(τk − τk) =j∑

k=1

(τπ (k) − τπ (k)

)

=j−1∑k=1

(k + 1)((

τπ (k) − τπ (k)

) − (τπ (k+1) − τπ (k+1)

))+ ( j + 1)

(τπ ( j) − τπ ( j)

) − (τπ (1) − τπ (1)

)=

j−1∑k=max(i,1)

(k + 1)((

τπ (k) − τπ (k+1)

) − (τπ (k) − τπ (k+1)

))+ ( j + 1)(τπ ( j) − τπ ( j)) − τπ (1) + τπ (1). (2.8)

3. Next, by expressing the summands on the r.h.s. of (2.8) in

terms of departure times of the original busy period we prove

that

n∑k=1

(vk −vk) = τn+1 − tn+1 −n+1∑k=1

sk + τπ (1). (2.9)

3.1 Let 0 < i = j ≤ n. Then (τπ ( j) − tn+1) − (τπ ( j) −tn+1) is just the reduction of vπ ( j) since tn+1 in the modi-

fied busy period compared to the original one as the portion

of the service capacity assigned to each of the requests in the

system is raised from 1/( j + 1) to 1/j until τn+1. Hence

τπ ( j) − τπ ( j) = − 1

j +1(τn+1 − tn+1).

Thus (2.8) provides

n∑k=1

(vk −vk) = τn+1 − tn+1 − τπ (1) + τπ (1). (2.10)

Since i > 0 implies τπ (1) ≥ τn+1, it follows that the original

busy period, which has length∑n+1

k=1 sk , ends at τπ (1), and

thus (2.10) yields (2.9).

3.2 Let 0 ≤ i < j ≤ n. Then (τπ ( j) − tn+1) − (τπ ( j) −tn+1) is just the reduction of vπ ( j) since tn+1 in the modi-

fied busy period compared to the original one as the portion

of the service capacity assigned to each of the requests is

raised from 1/( j + 1) to 1/j . Hence

τπ ( j) − τπ ( j) = − 1

j +1

(τπ ( j) − tn+1

). (2.11)

If i < k < j , then (τπ (k) − τπ (k+1)) − (τπ (k) − τπ (k+1)) is the

reduction of vπ (k) since τπ (k+1) as the portion of the ser-

vice capacity assigned to each of the requests is raised from

1/(k + 1) to 1/k. Hence(τπ (k) − τπ (k+1)

) − (τπ (k) − τπ (k+1)

)= − 1

k+1

(τπ (k) − τπ (k+1)

). (2.12)

Finally, because (τπ (i) − τπ (i+1)) − (τπ (i) − τπ (i+1)) is the re-

duction of vπ (i) since τπ (i+1) as the portion of the service

capacity assigned to each of the requests is raised from

1/(i + 1) to 1/ i until τn+1, it holds(τπ (i) − τπ (i+1)

) − (τπ (i) − τπ (i+1)

)= − 1

i +1

(τn+1 − τπ (i+1)

). (2.13)

Thus from (2.8) and (2.11)–(2.13) it follows

n∑k=1

(vk −vk) = I{i > 0}(τn+1 − τπ (1)

) − tn+1 + τπ (1).

(2.14)

If i = 0 then τn+1 > τπ (1), and therefore the original busy pe-

riod ends at τn+1 = ∑n+1k=1 sk , and hence (2.14) implies (2.9).

If i > 0 then τn+1 ≤ τπ (1), and the original busy period ends

at τπ (1) = ∑n+1k=1 sk , and hence (2.14) implies (2.9), too.

4. Since the modified busy period ends at τπ (1), from the work

conservation law it follows τπ (1) = ∑nk=1 sk , and hence from

(2.4), (2.9) and vn+1 = τn+1 − tn+1 we obtain

n+1∑k=1

(vk − sk) = 2n∑

k=1

(v∗k − sk) + τn+1 − tn+1 − sn+1

+ (vn+1 − sn+1)

= 2n∑

k=1

(v∗k − sk) + 2(vn+1 − sn+1)

= 2n+1∑k=1

(v∗k − sk),

where the last equality is valid as obviously v∗n+1 = vn+1.

�

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284 Queueing Syst (2006) 52: 281–286

From Lemma 2.1 we derive a relation between the sojourn

times of a finite busy period in a single server system under

PS and FCFS. As in Lemma 2.1 consider a sample path of

a finite busy period of n requests with service times sk , k =1, . . . , n. Additionally to the sojourn time vP

k (= vk) of the

k-th request under the PS discipline let vFk , k = 1, . . . , n, be

the corresponding sojourn time under the FCFS discipline.

Theorem 2.1. For the sojourn times vPk and vF

k of a finitebusy period of n requests in a single server system under PSand FCFS discipline, respectively, it holds

n∑k=1

(vP

k − sk) ≤ 2

n∑k=1

(vF

k − sk). (2.15)

In case of constant service times we have equality in (2.15).

Proof: Consider a sample path of a busy period of n re-

quests, and let 0 = t1 ≤ · · · ≤ tn be the arrival epochs of the

n requests. Analogously to Lemma 2.1 let v∗k be the sojourn

time of the k-th request in the PS system if the arrival process

is stopped after the arrival of the k-th request. Obviously, the

work conservation law provides

tk + v∗k ≤

k∑j=1

s j , k = 1, . . . , n, (2.16)

as the busy period induced by the first k arrivals

(t1, s1), . . . , (tk, sk) has length∑k

j=1 s j . If all service times

are equal, we have equality in (2.16) because then the re-

quests leave the system in the order of their arrivals. In case

of FCFS discipline it holds

tk + vFk =

k∑j=1

s j , k = 1, . . . , n. (2.17)

Combining (2.1) where vk = vPk , (2.16) and (2.17) we obtain

(2.15), where in (2.15) we have equality in case of constant

service times. �

Remark 2.1. The difference of the sojourn and service time

of a request can be interpreted as the waiting time of the

request also in case of a PS system, cf. [18] p. 107, namely

as the time which the request has to spend additionally to its

service time in the system in view of the presence of other

requests. Theorem 2.1 provides the sharp bound 2 in the worst

case comparative analysis for the sum of the waiting times

under PS and FCFS. Note that there is no finite bound for

the other direction because of the example (t1, s1) := (0, 2),

(t2, s2) := (1, 0).

3. Relations between the mean sojourn times forG/G/1 under PS and FCFS

In this section we derive some relations between the mean

stationary sojourn times for the single server system under

PS and FCFS with a stochastic arrival and service process.

By G/G/1 we denote the single server system where the

stochastic arrival and service process is a stationary and er-

godic sequence � = {[A�, S�]}∞�=−∞, where A� (≥0) denotes

the inter-arrival time between the �-th and (� + 1)–st request

and S� (≥0) the service requirement of the �-th request.3 If

additionally {S�}∞�=−∞ is a sequence of i.i.d. r.v.’s and inde-

pendent of the sequence {A�}∞�=−∞ then the system is de-

noted by G/G I/1. The arrival instants of the requests are

given by T� = − ∑−1k=� Ak , � < 0, T� = ∑�−1

k=0 Ak , � ≥ 0, cf.

e.g. [1, 7]. The marked point process � = {[T�, S�]}∞�=−∞(or equivalently the sequence �) is the arrival stationary

model of the input. By means of the inversion formula

for marked point processes, a corresponding stationary er-

godic marked point process � = {[T�, S�]}∞�=−∞ on R, with

· · · ≤ T−1 ≤ T0 ≤ 0 < T1 ≤ · · ·, can be given, describing the

time stationary version of the input, cf. e.g. [1, 7, 8]. (Note

that � has the Palm distribution of �.)

For the general G/G/1 system under any work conserving

discipline, in particular for the PS and FCFS discipline, the

stability condition reads

� := E S

E A< 1, (3.1)

where A and S denote generic r.v.’s of A� and S�, respectively,

cf. e.g. [1, 7]. Under (3.1) a uniquely determined stationary

workload process can be constructed (in the arrival as well as

time stationary model), having empty points and thus busy

periods of finite length. Hence for the general G/G/1 sys-

tem under PS a uniquely determined stationary process of

the number of requests and their residual service times can

be constructed along the lines of [7], but which will not be

outlined here.

By means of ergodicity arguments, cf. e.g. [1, 7, 8], the

sample path result of Theorem 2.1 yields a corresponding

one for the mean stationary sojourn times.

Theorem 3.1. Let � < 1. Then for the stationary sojourntimes V P of the G/G/1 − P S system and V F of the corre-sponding G/G/1 − FC F S system it holds

EV P −E S ≤ 2 (EV F −E S) (3.2)

3 Since A� and S� are assumed to be non-negative only, batch arrivalsas well as zero time requests are included.

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Queueing Syst (2006) 52: 281–286 285

with equality in case of deterministic service times, i.e., forthe G/D/1 system it holds

EV P −E S = 2 (EV F −E S). (3.3)

We will generalize (3.3) to a broader class of G/G I/1 sys-

tems, covering in particular corresponding known results for

G I/M/1 and M/G I/1. Note that for a G/G I/1 system in

the time stationary version � = {[T�, S�]}∞�=−∞ of the input,

{S�}∞�=−∞ is independent of {T�}∞�=−∞ and an i.i.d. sequence

of r.v.’s with the same distribution as {S�}∞�=−∞, cf. e.g. [7].

Consider the service time distributions Bi (x) := P(S ≤ x),

i ∈ {1, 2}:

B1(x) := 1 − p I{x <s}, x ∈ R+, (3.4)

B2(x) := 1 − p exp(−x/s), x ∈ R+, (3.5)

where p ∈ (0, 1] and s ∈ R+ \ {0}, i.e., Bi (x) is a mixture of

a zero and a deterministic or exponential time with mean s,

respectively.

Theorem 3.2. Let � < 1. Then for the systems G/G I/1 withG I = Bi , i ∈ {1, 2}, and M/G I/1 it holds

EV P − E S = 2(E S)2

E S2(EV F − E S), (3.6)

where V P and V F denote the stationary sojourn times in thecorresponding systems under the PS and FCFS discipline,respectively.

Proof: 1. For the M/G I/1 system (3.6) follows from the

well-known formulae for EV P and EV F , cf. e.g. [18] p. 109

and [17] p. 278.

2. Consider the systems G/G I/1 with G I = Bi , i ∈ {1, 2}.2.1 First let p = 1, i.e., we deal with the G/D/1 and

G/M/1 system, respectively. For a G/D/1 system (3.6) fol-

lows from (3.3) as E S2 = (E S)2. In case of a G/M/1 sys-

tem (3.6) is equivalent to EV P = EV F because of E S2 =2(E S)2. In view of the exponential service times, under the

PS as well as under the FCFS discipline the departure pro-

cess is a Poisson process of intensity 1/E S as long as there

are requests in the system. Consequently, the numbers of re-

quests in the system are stochastically equivalent under both

disciplines, and by Little’s formula it follows EV P = EV F .

2.2 Now let p ∈ (0, 1), i.e., the service times are a proper

mixture of a zero and a positive (deterministic or exponential)

time. Consider the time stationary input � = {[T�, S�]}∞�=−∞and the modified thinned input � = {[T�, S�]}∞�=−∞ with

· · · ≤ T−1 ≤ T0 ≤ 0 < T1 ≤ · · · consisting of all those re-

quests of � having a positive service time.4 In the following

let us endow all quantities and variables which are related

to � with a tilde, thus S denotes the generic deterministic

or exponential service time with mean s, and V P , V F are

the stationary sojourn times in the single server system with

input � under the PS and FCFS discipline, respectively. The

following observation is crucial: the requests with a zero

service time have under the PS discipline a sojourn time of

length zero whereas under the FCFS discipline they do not

have any impact on the workload process and hence on the

waiting times of the requests with positive service times.

This, (3.4), (3.5), and since the service times {S�}∞�=−∞ are

i.i.d. r.v.’s independent of the arrival process, yield in case of

G I = Bi , i ∈ {1, 2}, the relations

EV P = pEV P , EV F − E S = EV F − E S, (3.7)

E S = pE S, E S2 = pE S2. (3.8)

Note that EV P (EV F − E S) is the mean stationary so-

journ time (waiting time) in the corresponding G/D/1 − P S(FC F S) or G/M/1 − P S (FC F S) system, respectively,

where G stands for the process {T�}∞�=−∞ of arrival instants

of �, D for the constant service times s in case of B1(x) and

M for the exponential service times with mean s in case of

B2(x). (A rigorous proof of (3.7) can be given by applying

Palm’s formula for marked point processes and using the in-

dependence assumptions of the service times.) From (3.7),

(3.8) and step 2.1 above we obtain

EV P − E S = p(EV P − E S) = p2(E S)2

E S2(EV F − E S)

= 2(E S)2

E S2(EV F − E S).

�

Remark 3.1. For the G I/M/1 system the fact EV P = EV F

is well-known, cf. e.g. [13] p. 441. Theorem 3.2 states that

(3.6) holds for a much broader class than for G I/M/1 and

M/G I/1 systems.

Although Theorem 3.2 shows that (3.6) holds for a broad

class of G/G I/1 systems, this relation is not true for

G/G I/1 systems in general. Consider the MG I /G I/1 sys-

tem, i.e., batches of positive size X arrive according to a

Poisson process of intensity λ. Note that the MG I batch ar-

rival process is a special G process, cf. e.g. [1, 7]. The mean

sojourn time EV F of an arriving request under FCFS is given

4 The arrival stationary model of � can be given via Palm’s inversionformula for marked point processes, cf. e.g. [1, 7].

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286 Queueing Syst (2006) 52: 281–286

by, cf. e.g. [15] p. 277,

EV F = E S + E S

2(1 − �)

(�

E S2

(E S)2+ b

), (3.9)

where � = λ E X E S < 1, cf. (3.1), and b := E X2/E X −1 is the mean number of requests that arrive together with

the tagged arriving request other than this request. Since for

the MG I /G I/1 − P S system an explicit formula for EV P

seems not to be available, let us restrict to H2-distributed

service times with balanced means, i.e.,

H2(x) := p1(1 − exp(−x/s1))

+ p2(1 − exp(−x/s2)), x ∈ R+, (3.10)

where p1 = 1 − p2 ∈ (0, 1), s1, s2 ∈ R+ \ {0}, s1 �= s2 and

p1s1 = p2s2. (3.11)

Specializing the general results of [3, 10] for the

MG I /G I/1 − P S and MG I /G H/1 − P S system, where

G H stands for generalized hyperexponential distribution, re-

spectively, to the H2 distribution with balanced means, after

tedious algebra one obtains that for the MG I /H2/1 − P Ssystem with (3.11) it holds

EV P = E S + E S

2(1 − �)

(2� + b − ab

2 − �

), (3.12)

where a := (E S2 − 2(E S)2)/E S2 ∈ (0, 1). From (3.9),

(3.12) it follows

EV P − E S

= 2(E S)2

E S2

(1 + (1 − �)ab

(2 − �)(2� + (1 − a)b)

)(EV F − E S).

(3.13)

Therefore, for the MG I /H2/1 system with (3.11) it holds

EV P − E S ≥ 2(E S)2

E S2(EV F − E S) (3.14)

with equality iff b = 0, in which case we have a M/H2/1

system covered by Theorem 3.2.

Remark 3.2. As (3.6) holds for a broad class of G/G I/1

systems and implies (3.2), in case of the general G/G I/1

system (3.6) can be used as an approximation of EV P in

terms of EV F . The mean stationary sojourn time EV F un-

der FCFS can be approximated e.g. by a moment fitting of

the service time distribution B(x), preferentially by a two-

moment fitting to B1(x) as (3.6) is fulfilled for B1(x) and

B1(x) covers the full range of the first two moments. For the

G I/G I/1 system EV F can also be approximated by the for-

mula given in [11]. For a different approximation of EV P in

the G I/G I/1 system see [14].

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