steady-state analysis of a multi-class map/ph/c queue … outline background ... we consider a...

Steady-state Analysis of a Multi-class

MAP/PH/c Queue with Acyclic PH Retrials

Tugrul DayarDepartment of Computer Engineering

Bilkent [email protected]

(Joint Work with M. Can Orhan)To appear in Applied Probability Trust journal, 2016

14 January 2016

Outline

Outline

Background

Mathematical Model

Ergodicity Condition

KroneckerRepresentation

Numerical Results

Conclusion

TU-Dortmund-Informatik IV 14 January 2016 – 2 / 31

■ Background

▲ An Example

■ Mathematical Model

▲ An Example (cont’d)

■ Ergodicity Condition

▲ Necessary Part▲ Sufficient Part

■ Kronecker Representation

▲ An Example (cont’d)

■ Numerical Results■ Conclusion

Background

Outline

Background

An Example

Mathematical Model



Numerical Results

Conclusion


We consider a multi-class MAP/PH/c queueing system with

■ K customer classes■ class-dependent MAP (Markovian Arrival Process) arrivals■ class-dependent PH (phase-type) services in c servers■ class-dependent acyclic PH retrials of customers

who find the servers busy upon arrival

Model has infinite state space due to infinite retrial queue (orbit)

Background (cont’d)


Earlier Work on the Subject

■ Kulkarni: 2-class BM/G/1 with EXP retrial (JAP ’86)

■ Falin: Multi-class BM/G/1 with EXP retrial (AAP ’88)

■ Neuts, Rao: M/M/c with EXP retrial (QS ’90)

■ Falin, Templeton: Retrial Queues (Chapman and Hall, ’97)

■ Diamond, Alfa: MAP/PH/1 with EXP retrial (SM ’98)

■ Diamond, Alfa: M/PH/1 with PH retrial (EJOR ’99)

■ Choi, Chang: MAP1,2/M/c/∞,K with EXP retrial (MCM ’99)

■ Choi, Chang, Kim: MAP1,2/M/c/c+K,∞ with EXP retrial (TOP ’99)

■ He, Li, Zhao: BMAP/PH/c/c+K with PH retrial, impatience (QS ’00)

■ Dudin, Klimenok: BMAP/SM/1 with EXP retrial (QS ’00)

■ Breuer, Dudin, Klimenok: BMAP/PH/c with EXP retrial (QS ’02)

■ Artalejo, Gomez-Corral: MAP/PH/c with PH retrial, finite source (IEEE CL ’07)



Earlier Work on the Subject

■ Artalejo, Gomez-Corral: Retrial Queueing Systems (Springer ’08)

■ Shin: M/M/c with PH2 retrial (JAMI ’11)

■ Shin, Moon: M/M/c with PH retrial (EJOR ’11)

■ Phung-Duc, Kawanishi: M/M/c/c+K with EXP retrial, after-call work (NACO ’11)

■ Kim, Mushko, Dudin: BMAP/PH/c with EXP retrial, impatience (AOR ’12)

■ Artalejo, Phung-Duc: M/M/c with EXP retrial, two-way communication (JIMO ’12)

■ Kumar, Sohraby, Kim: M/M/c with PH retrial, impatience (IEEE CL ’13)

■ Chakravarty: MAP/PH/c with PH retrial (Ed. book, Springer ’13)

■ Shin, Moon: Multiclass M/M/c with EXP retrial (MCAP ’14)

■ Phung-Duc, Kawanishi: M/M/c/c+K with EXP retrial, after-call work, impatience(PE ’14)

■ Sakurai, Phung-Duc: M/M/c with EXP retrial, two-way communication,class-dependent service (TOP ’15)


Outline

Background

An Example

Mathematical Model



Numerical Results

Conclusion


Customers of class k ∈ {1, . . . ,K} arrive according to MAP

(Ck,Dk) of order mAk

which is an irreducible MC with state space {1, . . . ,mAk } and

generator matrix (Ck +Dk), whereCk: nonsingular with negative diagonal,

nonnegative off-diagonal elements, andcharacterizing transitions without an arrival

Dk: nonnegative and characterizing transitions with one arrival

There exists a nonnegative row vector θk ∈ R1×mA

k

≥0 s.t.

θk(Ck +Dk) = 0 and θke = 1

Average arrival rate of class k: λk = θkDke


Outline

Background

An Example

Mathematical Model



Numerical Results

Conclusion


Customers of class k ∈ {1, . . . ,K} receive service according toPH distribution

(βk,T k) of order mSk and T 0

k = −T ke

which is distribution of time until absorption in a finite MC with

■ state space {1, . . . ,mSk + 1}

■ generator matrix Tk =

[

T k T 0k

0 0

]

(mSk+1)×(mS

k+1)

■ initial probability vector (βk, 1− βke) ∈ R1×(m+1)≥0

whereT k: nonsingular with negative diagonal and

nonnegative off-diagonal elements

Average service rate of class k: µk = [−βk(T k)−1e]−1


Outline

Background

An Example

Mathematical Model



Numerical Results

Conclusion


If all servers are busy upon arrival, an arriving customer of class kjoins orbit k and retries to capture a server according to acyclic

PH distribution

(ξk,Uk) of order mRk and U0

k = −Uke

where Uk is upper-triangular

Average retrial rate of class k: δk = [−ξk(Uk)−1e]−1

An Example

Outline

Background

An Example

Mathematical Model



Numerical Results

Conclusion


Let K = 2, c = 2, and

C1 =

[

−0.8 0.8

0 −0.8

]

2×2

, D1 =

[

0 0

0.8 0

]

2×2

, C2 =[

−0.3]

1×1, D2 =

[

0.3]

1×1

ξ1=

[

1 0]

1×2, U1 =

[

−1 1

0 −1

]

2×2

, ξ2=

[

1]

1×1, U2 =

[

−0.5]

1×1

β1=

[

0.75 0.25]

1×2, T 1 =

[

−1 0.25

0 −0.25

]

2×2

, β2=

[

1]

1×1, T 2 =

[

−0.5]

1×1

Hence,

U0

1 =

[

0

1

]

2×1

, U0

2 =[

0.5]

1×1, T

0

1 =

[

0.75

0.25

]

2×1

, T0

2 =[

0.5]

1×1

and

λ1 = 0.4, λ2 = 0.3, δ1 = δ2 = 0.5, µ1 = 0.4, µ2 = 0.5

mA1 = 2, m

A2 = 1, m

R1 = 2, m

R2 = 1, m

S1 = 2, m

S2 = 1

Mathematical Model

Outline

Background

Mathematical Model

An Example (cont’d)



Numerical Results

Conclusion


Let

■ Xk(t): phase of arrival process of class k customers■ XbR

k+iR

k(t): # of class k retrial customers in phase iRk

■ XbSk+iS

k(t): # of servers serving class k customers in phase iSk

for iRk = 1, . . . ,mRk and iSk = 1, . . . ,mS

k , where

mR =

K∑

k=1

mRk , mS =

K∑

k=1

mSk

bRk = K +k−1∑

k′=1

mRk′ , bSk = K +mR +

k−1∑

k′=1

mSk′

Then we have multi-dimensional MC

X(t) = {X1(t), . . . , XK+mR+mS (t) : t ≥ 0}

Mathematical Model (cont’d)

Outline

Background

Mathematical Model




Numerical Results

Conclusion


with state space S = SA × SR × SS where

SA = ×Kk=1

{

1, . . . ,mAk

}

SR = ZmR

≥0

SS = {y = (y1, . . . , ymS ) ∈ ZmS

≥0 | ye ≤ c}

and a possible state representation of the model is

x = (x1, . . . , xK+mR+mS ) ∈ S

# of busy servers in state x: n(x) =∑K

k=1

∑mSk

i=1 xbSk+i

SS is defined so that n(x) ≤ c

|SS | =c

∑

i=0

(i+mS − 1)!

i! (mS − 1)!


Outline

Background

Mathematical Model




Numerical Results

Conclusion


The model has 8 dimensionssince K = 2, mR = 3, mS = 3

Moreover, bR1 = 2, bR2 = 4, bS1 = 5, bS2 = 7, n(x) = x6 + x7 + x8

Therefore, the state space of the MC is given by

S = SA × SR × SS ,

whereSA = {1, 2} × {1} , SR = Z

3≥0

SS = {(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), (0, 1, 1),

(0, 2, 0), (1, 0, 0), (1, 0, 1), (1, 1, 0), (2, 0, 0)}

for c = 2



Matrix QAk is associated with arrival of class k customers:

QAk (x,y) =

Ck(xk, i) if i 6= xk and y = x+ (i− xk)eTk

Dk(xk, i)ξk(j) if n(x) = c and y = x+ (i− xk)eTk + eT

bRk+j

Dk(xk, i)βk(j′) if n(x) < c and y = x+ (i− xk)e

Tk + eT

bSk+j′

0 otherwise

for i = 1, . . . ,mAk , j = 1, . . . ,mR

k , j′ = 1, . . . ,mS

k , and x,y ∈ S

Matrix QRk is associated with retrial of class k customers:

QRk (x,y) =

xbRk+iUk(i, j) if i 6= j and y = x− eT

bRk+i

+ eTbRk+j

xbRk+iU

0k(i)βk(j

′) if n(x) < c and y = x− eTbRk+i

+ eTbSk+j′

0 otherwise

for i, j = 1, . . . ,mRk , j

′ = 1, . . . ,mSk , and x,y ∈ S



Matrix QSk is associated with service of class k customers:

QSk (x,y) =

xbSk+iT k(i, j) if i 6= j and y = x− eT

bSk+i

+ eTbSk+j

xbSk+iT

0k(i) if y = x− eT

bSk+i

0 otherwise

for i, j = 1, . . . ,mSk and x,y ∈ S

Therefore, generator matrix underlying X(t) can be written as

Q = Qoff + diag(−Qoffe) with Qoff =∑K

k=1

(

QAk +QR

k +QSk

)


Outline

Background

Mathematical Model


Necessary Part

Sufficient Part


Numerical Results

Conclusion


With the help of Lyapunov functions, we show that

∑Kk=1

λk

µk< c

is a necessary and sufficient ergodicity condition

Lyapunov function used to show sufficiency aids intruncating S with a given steady-state probability mass

We start with Lyapunov functions that work for simple models,and add terms for additional complexities of model

Two vectors, uk and vk, used in additional terms are introduced

Ergodicity Condition (cont’d)

Outline

Background

Mathematical Model


Necessary Part

Sufficient Part


Numerical Results

Conclusion


There exists a unique vector uk ∈ RmA

k×1 for MAP (Ck,Dk)

and λk = θkDke s.t.

(Ck +Dk)uk = λke−Dke and uke = 1

Since transition rates describing MAP arrivals depend on phase,elements of uk are used in additional terms to obtain a conditionbased on λk instead of phase-dependent arrival rates in Dk

There exists a unique vector vk ∈ RmS

k×1

≥0 for PH service

distribution (βk,T k) and µk = [−βk(T k)−1e]−1 s.t.

vk = −µk(T k)−1e and βkvk = 1

Since transition rates describing PH service depend on phase,elements of vk are used in additional terms to obtain a conditionbased on µk instead of phase-dependent service rates in T 0

k

Necessary Part

Outline

Background

Mathematical Model


Necessary Part

Sufficient Part


Numerical Results

Conclusion


(Asmussen ’03; Fayolle, Malyshev, Menshikov ’95)MC with generator matrix Q is non-ergodic if there exists twoconstants τ, σ ∈ R and a Lyapunov function f : S → R s.t.(i)

∑

y∈S P (x,y)|f(y)− f(x)| ≤ τ for x ∈ S and

(ii)∑

y∈S P (x,y) (f(y)− f(x)) ≥ 0 for x ∈ R,

where R = {x ∈ S | f(x) > σ} and (S \ R) are non-empty, and

matrix P ∈ R|S|×|S|≥0 :

P (x,y) =

{

Q(x,y)/|Q(x,x)| if y 6= x

0 otherwisefor x,y ∈ S

Inspired by (Falin, Templeton ’97), we use

f(x) =∑K

k=11µk

(

∑mRk

i=1 xbRk +i

)

+∑K

k=1uk(xk)

µk

+∑K

k=11µk

(

∑mSk

i=1 vk(i)xbSk+i

)

Sufficient Part

Outline

Background

Mathematical Model


Necessary Part

Sufficient Part


Numerical Results

Conclusion


(Tweedie ’75) MC is ergodic if and only if there exists aLyapunov function g : S → R and a finite set C ⊂ S s.t.(i) {x ∈ S | g(x) ≤ τ} is finite for all τ < ∞,(ii) d(x) ≤ −γ for all x ∈ S \ C and some γ > 0, and(iii) d(x) < ∞ for all x ∈ S,

where d(x) =∑

y∈S Q(x,y) (g(y)− g(x)) is called the drift instate x ∈ S

For acyclic PH retrial distribution (ξk,Uk),

let Uk = Uk + diag(U0k) and ηk ∈ R

mRk×1 be given as

ηk(i) =

{

−c/µk if i ∈ Ik0 otherwise

for i = 1, . . . ,mRk ,

where Ik = {i ∈ {1, . . . ,mRk } | U0

k(i) = 0}

Then there exists wk ∈ RmR

k×1 s.t. Ukwk = ηk

Sufficient Part (cont’d)


There exist infinitely many solutions to Ukwk = ηk. In this work,we set wk(i) to 1 if row i of Uk is zero and use

g(x) =∑K

k′=1

∑Kl′=1

12µk′µl′

(

∑mRk′

i=1 xbRk′+i

)(

∑mRl′

i=1 xbRl′+i

)

+∑K

k′=1

(

∑Kl′=1

ul′ (xl′ )µk′µl′

)

(

∑mRk′

i=1 xbRk′+i

)

+∑K

k′=1

(

∑Kl′=1

1µk′µl′

(

∑mSl′

j=1 vl′(j)xbSl′+j

))(

∑mRk′

i=1 xbRk′+i

)

+∑K

k′=1

∑mRk′

i=1 wk′(i)xbRk′+i +

∑Kk′=1 αk′(x)

(

∑mSk′

i=1 xbSk′+i

)

,

where ak(i,x) = wk(i) +1

2µ2

k

+∑K

k′=1uk′ (xk′ )µkµk′

+∑K

k′=11

µkµk′

(

∑mSk′

j′=1 vk′(j′)xbS

k′+j′

)

for i 6∈ Ik,

αk(x) = mini 6∈Ik(

ak(i,x)− c/(U0k(i)µk)

)

for x ∈ S, k = 1, . . . ,K

Kronecker Representation

Outline

Background

Mathematical Model




Numerical Results

Conclusion


Truncated model in which size of retrial phase i of orbit k is

Nk,i = max{xbRk+i | x = (x1, . . . , xK+mR+mS ) ∈ C}+ 1

for i = 1, . . . ,mRk , k = 1, . . . ,K

Truncated state space is S = SA × SR × SS

where SR = ×Kk=1

(

×mR

k

i=1{0, . . . , Nk,i − 1})

Then multi-dimensional MC X(t) with finite state space S andgenerator matrix Q:

Q(x,y) =

{

Q(x,y) if x 6= y

−∑

y∈S Q(x,y) otherwisefor x,y ∈ S

Kronecker Representation (cont’d)

Outline

Background

Mathematical Model




Numerical Results

Conclusion


Let L = |SS | and ϕ : SS → {1, . . . , L} be given by

ϕ(xS) =mS−1∑

h=1

xSh−1

∑

i=0

c−rh−i∑

j=0

(j +mS − h− 1)!

j! (mS − h− 1)!+ xSmS + 1,

where rh =∑h−1

h′=1 xSh′ for xS = (xS1 , . . . , x

SmS ) ∈ SS

ϕ(xS): lexicographical order of state xS in SS

ϕ(xS) is one-to-one and ontoHence, ϕ−1 : {1, . . . , L} → SS is well-defined

Let Sl = SA × SR × {ϕ−1(l)} for l = 1, . . . , LThen S1, . . . , SL is a partitioning of S; that is,⋃L

l=1 Sl = S and Sl ∩ Sl′ = ∅ for l 6= l′ and l, l′ = 1, . . . , L



The truncated generator matrix is

Q = Qoff

+ diag(−Qoff

e)

where

Qoff

=

Qoff1,1 . . . Q

off1,L

.... . .

...

QoffL,1 . . . Q

offL,L

Let auxiliary matrices with transitions among subsystems corresponding to phasesof arrival processes be

W (C,k′) = (⊗k′−1

k=1 ImAk) ⊗C

offk′ ⊗ (

⊗Kk=k′+1 ImA

k)

W (D,k′) = (⊗k′−1

k=1 ImAk) ⊗Dk′ ⊗ (

⊗Kk=k′+1 ImA

k)

for k′ = 1, . . . ,K, where matrix Coffk′ includes only off-diagonal elements of Ck′


Outline

Background

Mathematical Model




Numerical Results

Conclusion


For n ∈ Z≥0, let

E−n (i, j) =

{

i if j = i− 10 otherwise

, E+n (i, j) =

{

1 if j = i+ 10 otherwise

for i, j = 0, . . . , n− 1

Then auxiliary matrices with transitions among subsystemscorresponding to phases of retrial processes are

A(k′,i′) =K⊗

k=1

mRk

⊗

h=1

A(k′,i′)k,h , R(k′,i′,j′) =

K⊗

k=1

mRk

⊗

h=1

R(k′,i′,j′)k,h

S(k′,i′) =

K⊗

k=1

mRk

⊗

h=1

S(k′,i′)k,h ,


Outline

Background

Mathematical Model




Numerical Results

Conclusion


where

A(k′,i′)k,h =

{

E+Nk,h

if k = k′ and h = i′

INk,hotherwise

R(k′,i′,j′)k,h =

E−Nk,h


E+Nk,h

if k = k′ and h = j′

INk,hotherwise

S(k′,i′)k,h =

{

E−Nk,h


INk,hotherwise

for i′, j′, h = 1, . . . ,mRk , i′ 6= j′, k, k′ = 1, . . . ,K

T. Dayar, M. C. Orhan, On vector-Kronecker productmultiplication with rectangular factors, SISC 37(5), 2015.



Then Qoffl,l′ =

∑Kk=1(W

(C,k) ⊗ IN )

+ 1n(l)=c

∑Kk=1

∑mRk

i=1 ξk(i)(W(D,k) ⊗A(k,i))

+∑K

k=1

∑mRk

i=1

∑mRk

j=1j 6=i

Uk(i, j)(ImR ⊗R(k,i,j)) if sl,l′ = 0

1n(l)<cβk(j)(W(D,k) ⊗ IN )

+ 1n(l)<c

∑mRk

i=1U0k(i)βk(j)(ImR ⊗ S(k,i)) if sl,l′ = ebS

k+j

xSbSk+iT k(i, j)(ImR ⊗ IN ) if (sl,l′ = −ebS

k+i + ebS

k+j

and i 6= j)xSbSk+iT 0

k(i)(ImR ⊗ IN ) if sl,l′ = −ebSk+i

0 otherwise

,

wheresl,l′ = ϕ−1(l′)− ϕ−1(l), xS = ϕ−1(l), n(l) = xSe,

bSk =∑k−1

k′=1mSk′ , N =

∏Kk=1

∏mRk

i=1Nk,i


Outline

Background

Mathematical Model




Numerical Results

Conclusion


|SS | = 10, N1,1 = 97, N1,2 = 111, N2,1 = 139ϕ(0, 0, 0) = 1, ϕ(0, 0, 1) = 2, ϕ(0, 0, 2) = 3, ϕ(0, 1, 0) = 4ϕ(0, 1, 1) = 5, ϕ(0, 2, 0) = 6, ϕ(1, 0, 0) = 7, ϕ(1, 0, 1) = 8ϕ(1, 1, 0) = 9, ϕ(2, 0, 0) = 10

Hence, Qoff

includes 100 blocks and the auxiliary matrices are

Coff1 =

[

0 0.80 0

]

2×2

, Coff2 = [0]1×1, A(1,1) = E+

N1,1⊗IN1,2

⊗IN2,1

A(1,2) = IN1,1⊗E+

N1,2⊗ IN2,1

, A(2,1) = IN1,1⊗ IN1,2

⊗E+N2,1

R(1,1,2) = E−N1,1

⊗E+N1,2

⊗IN2,1, R(1,2,1) = E+

N1,1⊗E−

N1,2⊗IN2,1

S(1,1) = E−N1,1

⊗ IN1,2⊗ IN2,1

, S(1,2) = IN1,1⊗E−

N1,2⊗ IN2,1

S(2,1) = IN1,1⊗ IN1,2

⊗E−N2,1

, W (C,1) = Coff1 ⊗ I1

W (C,2) = I2 ⊗Coff2 , W (D,1) = D1 ⊗ I1, W (D,2) = I2 ⊗D2



The nonzero blocks of Qoff

are

Q1,1 = Q2,2 = Q4,4 = Q7,7 = (W (C,1) ⊗ IN ) + (I2 ⊗R(1,1,2))

Q3,3 = Q5,5 = Q6,6 = Q8,8 = Q9,9 = Q10,10

= (W (C,1) ⊗ IN ) + (W (D,1) ⊗A(1,1)) + (W (D,2) ⊗A(2,1)) + (I2 ⊗R(1,1,2))

Q1,2 = Q2,3 = Q4,5 = Q7,8 = (W (D,2) ⊗ IN ) + 0.5(I2 ⊗ S(2,1))

Q1,4 = Q2,5 = Q4,6 = Q7,9 = 0.25(W (D,1) ⊗ IN ) + 0.25(I2 ⊗ S(1,2))

Q1,7 = Q2,8 = Q4,9 = Q7,10 = 0.75(W (D,1) ⊗ IN ) + 0.75(I2 ⊗ S(1,2))

Q4,1 = Q5,2 = Q7,4 = Q8,5 = Q9,6 = Q9,7 = 0.25I2N

Q7,1 = Q8,2 = Q9,4 = 0.75I2N

Q2,1 = Q5,4 = Q6,4 = Q8,7 = Q10,9 = 0.5I2N , Q3,2 = I2N , Q10,7 = 1.5I2N

Numerical Results

Outline

Background

Mathematical Model



Numerical Results

Conclusion


Kronecker solver built on the Nsolve package of the APNNtoolbox to

■ obtain truncated state space of the model■ generate Kronecker structured matrix of truncated model■ compute steady-state solution using SOR with ω = 0.9

||πQ||∞ < 10−15

Six different models with ǫ = 0.2

ERL1: model introduced in our exampleERL2: (0.75C1, 0.75D1)ERL3: (0.5C1, 0.5D1)

EXP1: ERL1 except retrial time of class 1 customers ≈ exp(0.5)EXP2: EXP1 except (0.75C1, 0.75D1)EXP3: EXP1 except (0.5C1, 0.5D1)

Numerical Results (cont’d)


Traffic intensity: ρ = (∑K

k=1 λk/µk)c−1

|S|: # of states in truncated state space

E1,1: average # of class 1 customers in retrial phase 1E1,2: average # of class 1 customers in retrial phase 2E2,1: average # of class 2 customers in retrial phase 1

Pblock: probability of finding all servers busy

||πQ||1: 1-norm of the residual vector of the truncated model||πQ||1: 1-norm of the residual vector of the infinite model

||πQ||1 =∑

x∈S

|r(x)−∑

y 6∈S

π(x)Q(x,y)|+∑

x∈S

∑

y 6∈S

π(x)Q(x,y) with r = πQ

Numerical Results (cont’d)


λ1/µ1 > λ2/µ2 and δ1 = δ2

Hence, # in orbit 1 increases faster than # in orbit 2

Table 1: Numerical resultsModel ρ |S| E1,1 E1,2 E2,1 Pblock ||πQ||1 ||πQ||1ERL1 0.8 29, 932, 260 0.2584 2.0165 2.8985 0.6784 9e− 14 9e− 14ERL2 0.6 811, 800 0.1097 0.4422 0.6756 0.4164 2e− 14 2e− 12ERL3 0.4 12, 800 0.0317 0.0860 0.1511 0.2048 6e− 15 7e− 6EXP1 0.8 270, 400 3.1663 2.5486 0.6755 8e− 14 8e− 14EXP2 0.6 25, 600 0.7154 0.6302 0.4142 2e− 14 2e− 11EXP3 0.4 1, 600 0.1464 0.1461 0.2038 4e− 15 6e− 6

Conclusion

Outline

Background

Mathematical Model



Numerical Results

Conclusion


In ERL1, ERL2, and ERL3,E1,2/E1,1 is larger when Pblock is larger

# in orbit 1 of ERLi < # in orbit 1 of EXPi for i = 1, 2, 3since customers in phase 1 in ERLi are not blockedif they retry when all servers are busy

Because servers are less likely to be captured by class 1 customersin EXPi, # in orbit 2 of ERLi is larger than that of EXPi

Relative difference between average numbers of retrial customersin ERLi and EXPi becomes relatively large as ρ increases.

Relative difference between blocking probabilities is around 0.005.

In general, truncation error is larger than numerical error.

As ρ increases, choosing a smaller ǫ value does not introduceadditional inaccuracy to the computed solution

steady-state analysis of a multi-class map/ph/c queue … outline background ... we consider a...

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