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CTMC Analysis for CAS+Q

6.7 System Analytical Models and Assumptions

6.7.1 CTMC Analysis for CAS+Q

Possible states of the system is represented asαΆ‘ = {πœ™ βŽΈπœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2 β‰₯ 0; 𝑧(πœ™) ≀ 𝑀 βˆ— 𝑆, 𝑄1≀ 𝑄1π‘šπ‘Žπ‘₯, 𝑄2≀ 𝑄2π‘šπ‘Žπ‘₯; βˆ‘ 𝑉𝑖=1𝑛 πœƒπ‘›π‘šπ‘–π‘›π‘– < πœƒπ‘›π‘šπ‘–π‘›π‘– , if 𝑄2> 0; βˆ‘ 𝑉𝑖=1𝑛 πœƒπ‘›π‘– < πœƒπ‘› , if 𝑄1> 0}. The steady state probabilities πœ‹(πœ™) can be calculated with the balance and the normalized equation as shown in Equation (6.7) where ψ is the transition rate matrix [8].

π𝛙 = 0, βˆ‘ πœ‹(πœ™) = 1 (6.7)

πœ™βˆˆ αΆ‘

Subsequently, the summary of all the possible state transitions is found in the tables below. The transition tables captures the varying wireless channel conditions and three event pairs of PU/SU (departure and arrival) which are; PU arrival/departure, π‘†π‘ˆπ‘Ž and π‘†π‘ˆπ‘ arrival/departure respectively.

137 Table 6.1

External transitions from present state πœ™ = {πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2} of CAS+Q [9]

A PU Departure

Table 6.2

External transitions from present state πœ™ = {πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2} of CAS+Q A π‘†π‘ˆπ‘ / π‘†π‘ˆπ‘Ž Arrivals

No .

Next state Possible State Events Transition Rate

PU rate PU Departure

(1) ( πœ” βˆ’ 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU departure irrespective of the SUs channels condition.

πœ”πœ‡π‘

PU departure is independent of the varying channel condition of the SUs

No .

Next state Possible State Events SU

Transition Rate π‘†π‘ˆπ‘ Arrival

[1] (πœ”, 𝑉𝑔(π‘Ž) . . , . . 𝑉𝑔(𝑏)+ 1, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) A π‘†π‘ˆπ‘ in good state arrivals. Sufficient idle channel-slots available, with the likelihood of being admitted, If the number of idle slots (πœƒπ‘ π‘’) available is greater than or equal to the summation of channel-slots required by 𝑉𝑔 i.e. If πœƒπ‘ π‘’β‰₯ βˆ‘π‘‰π‘–=1𝑔 πœƒπ‘”π‘šπ‘–π‘›π‘– , βˆ€ 𝑧(πœ™) < 𝑀 βˆ— 𝑆 as in equation (6.3)

[πœƒπ‘”π‘šπ‘–π‘›π‘‰π‘”(𝑏) 𝑧(πœ™) ] πœ†π‘ 

Otherwise, it is blocked which implies NOT admitted i.e. if πœƒπ‘ π‘’β‰€

βˆ‘π‘‰π‘–=1𝑔 πœƒπ‘”π‘šπ‘–π‘›π‘– , βˆ€{ 𝑀 βˆ— 𝑆 βˆ’ 𝑧(πœ™)} <

πœƒπ‘”π‘šπ‘–π‘› as in equation (6.4)

[2] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž). . , π‘‰π‘š(𝑏)+ 1, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) A π‘†π‘ˆπ‘ in moderate state arrivals.

Sufficient idle channel-slots available, with the likelihood of being admitted, If the number of idle slots ( πœƒπ‘ π‘’) available is greater than or equal to the summation of channel-slots required by π‘‰π‘š i.e. If πœƒπ‘ π‘’β‰₯ βˆ‘π‘‰π‘–=1π‘š πœƒπ‘šπ‘šπ‘–π‘›π‘– , βˆ€ 𝑧(πœ™) <

𝑀 βˆ— 𝑆 as in equation (6.3)

[πœƒπ‘šπ‘šπ‘–π‘›π‘‰π‘š(𝑏) 𝑧(πœ™) ] πœ†π‘ 

Otherwise, it is blocked which implies NOT admitted i.e. if πœƒπ‘ π‘’<

138

βˆ‘π‘‰π‘–=1π‘š πœƒπ‘šπ‘šπ‘–π‘›π‘– , βˆ€{ 𝑀 βˆ— 𝑆 βˆ’ 𝑧(πœ™)} <

πœƒπ‘šπ‘šπ‘–π‘› as in equation (6.4)

[3] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž). . 𝑉𝑏(𝑏)+ 1, 𝑄1, 𝑄2) A π‘†π‘ˆπ‘ in bad state arrivals. Sufficient idle channel-slots available, with the likelihood of being admitted, If the number of idle slots (πœƒπ‘ π‘’) available is greater than or equal to the summation of channel-slots required by 𝑉𝑏 i.e. If πœƒπ‘ π‘’β‰₯ βˆ‘π‘‰π‘–=1𝑏 πœƒπ‘π‘šπ‘–π‘›π‘– , βˆ€ 𝑧(πœ™) < 𝑀 βˆ— 𝑆 as in equation (6.3)

. [πœƒπ‘π‘šπ‘–π‘›π‘‰π‘(𝑏)

𝑧(πœ™) ] πœ†π‘ 

Otherwise, it is blocked which implies NOT admitted i.e. if πœƒπ‘ π‘’β‰€

βˆ‘π‘‰π‘–=1𝑏 πœƒπ‘π‘šπ‘–π‘›π‘– , βˆ€{ 𝑀 βˆ— 𝑆 βˆ’ 𝑧(πœ™)} <

πœƒπ‘π‘šπ‘–π‘› as in equation (6.4).

[4]

(πœ”, 𝑉𝑔(π‘Ž). . , 𝑉𝑔(𝑏)

+ 1, . , 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯βˆ’. , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2)

π‘†π‘ˆπ‘ arrivals in good state. A π‘†π‘ˆπ‘with maximum no. of channel-slots(πœƒπ‘”π‘šπ‘Žπ‘₯) donates minimum no. of channel-slots (πœƒπ‘”π‘šπ‘–π‘›) ….

πœ†π‘ 

[5] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)… , π‘‰π‘š(𝑏)

+ 1, . , π‘‰π‘šπ‘šπ‘Žπ‘₯βˆ’(𝑏) , 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2)

π‘†π‘ˆπ‘ arrivals in moderate state. A π‘†π‘ˆπ‘ with maximum no. of channel- slots(πœƒπ‘šπ‘šπ‘Žπ‘₯) donates minimum no. of channel-slots (πœƒπ‘šπ‘šπ‘–π‘›)….

πœ†π‘ 

[6] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž). . , 𝑉𝑏(𝑏)

+ 1, 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑄1, 𝑄2)

π‘†π‘ˆπ‘ arrivals in bad state. A π‘†π‘ˆπ‘with maximum no. of channel-slots(πœƒπ‘π‘šπ‘Žπ‘₯) donates minimum no. of channel-slots (πœƒπ‘π‘šπ‘–π‘›)….

πœ†π‘ 

[7] (πœ”, 𝑉𝑔(π‘Ž). . , 𝑉𝑔(𝑏)+ 1, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘ arrivals in good state. New π‘†π‘ˆπ‘ call/request is queued in 𝑄2

same as (4) [8] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž). . , π‘‰π‘š(𝑏)+ 1, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘ arrivals in moderate state. New

π‘†π‘ˆπ‘ call/request is queued in 𝑄2

same as (5) [9] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž). . , π‘‰π‘š(𝑏)+ 1, 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘ arrivals in bad state. New π‘†π‘ˆπ‘

call/request is queued in 𝑄2…. same as (6) 𝑺𝑼𝒂 Arrival

[10] (πœ”, 𝑉𝑔(π‘Ž)+ 1. . , . 𝑉𝑔(𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) An π‘†π‘ˆπ‘Ž in good state Arrivals.

Sufficient idle channel-slots available, with the likelihood of being admitted, If the number of idle slots ( πœƒπ‘ π‘’) available is greater than or equal to the summation of channel-slots required by 𝑉𝑔 i.e. If πœƒπ‘ π‘’β‰₯ βˆ‘π‘‰π‘–=1𝑔 πœƒπ‘”π‘– , βˆ€ 𝑧(πœ™) <

𝑀 βˆ— 𝑆 as in equation (5.4) ….

[πœƒπ‘”π‘‰π‘”(π‘Ž) 𝑧(πœ™) ] πœ†π‘ 

Otherwise, it is blocked which implies NOT admitted i.e. if πœƒπ‘ π‘’β‰€

βˆ‘π‘‰π‘–=1𝑔 πœƒπ‘”π‘– ,

{ 𝑀 βˆ— 𝑆 βˆ’ 𝑧(πœ™)} < πœƒπ‘” as in equation (6.2)

[11] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)+ 1. . , π‘‰π‘š(𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) A π‘†π‘ˆπ‘Ž in moderate state arrivals.

Sufficient idle channel-slots available, with the likelihood of being admitted,

139

If the number of idle slots ( πœƒπ‘ π‘’) available is greater than or equal to the summation of channel-slots required by π‘‰π‘š i.e.

If πœƒπ‘ π‘’β‰₯ βˆ‘π‘‰π‘–=1π‘š πœƒπ‘šπ‘– , βˆ€ 𝑧(πœ™) < 𝑀 βˆ— 𝑆 as in equation (5.4)

[πœƒπ‘šπ‘‰π‘š(π‘Ž) 𝑧(πœ™) ] πœ†π‘ 

Otherwise, it is blocked which implies NOT admitted i.e. if πœƒπ‘ π‘’β‰€

βˆ‘π‘‰π‘–=1π‘š πœƒπ‘šπ‘– , βˆ€{ 𝑀 βˆ— 𝑆 βˆ’ 𝑧(πœ™)} < πœƒπ‘š as in equation (6.2)

[12] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)+ 1, . . 𝑉𝑏(π‘Ž), 𝑄1, 𝑄2) A π‘†π‘ˆπ‘Ž in bad state arrivals. Sufficient idle channel-slots available, with the likelihood of being admitted, If the number of idle slots (πœƒπ‘ π‘’) available is greater than or equal to the summation of channel-slots required by 𝑉𝑏 i.e. If πœƒπ‘ π‘’β‰₯ βˆ‘π‘‰π‘–=1𝑏 πœƒπ‘π‘– , βˆ€ 𝑧(πœ™) < 𝑀 βˆ— 𝑆 as in equation (5.4)

. [πœƒπ‘π‘‰π‘(π‘Ž)

𝑧(πœ™) ] πœ†π‘ 

Otherwise, it is blocked which implies NOT admitted i.e. if πœƒπ‘ π‘’β‰€

βˆ‘π‘‰π‘–=1𝑏 πœƒπ‘π‘– , βˆ€{ 𝑀 βˆ— 𝑆 βˆ’ 𝑧(πœ™)} < πœƒπ‘ as in equation (6.1)

[13] (πœ”, 𝑉𝑔(π‘Ž)+ 1, . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯βˆ’, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž arrivals in good state. A π‘†π‘ˆπ‘ with πœƒπ‘”π‘šπ‘Žπ‘₯ donates πœƒπ‘” channel-slots.

πœ†π‘  [14] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)+ 1, . . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž arrivals in moderate state. An

π‘†π‘ˆπ‘with πœƒπ‘šπ‘šπ‘Žπ‘₯ donates πœƒπ‘š channel- slots

πœ†π‘ 

[15] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)+ 1, . . 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž arrivals in good state. A π‘†π‘ˆπ‘with πœƒπ‘π‘šπ‘Žπ‘₯ donates πœƒπ‘ channel-slots…

πœ†π‘  [16] (πœ”, 𝑉𝑔(π‘Ž)+ 1, . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯βˆ’, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1

βˆ’ 1, 𝑄2)

π‘†π‘ˆπ‘Ž arrivals in good state. New π‘†π‘ˆπ‘Ž call/request is queued in 𝑄1

same as (4) [17] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)+ 1, . . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑉𝑏(π‘Ž,𝑏), 𝑄1

βˆ’ 1, 𝑄2)

π‘†π‘ˆπ‘Ž arrivals in moderate state. New π‘†π‘ˆπ‘Ž call/request is queued in 𝑄1

same as (5) [18] (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)+ 1, . . 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑄1

βˆ’ 1, 𝑄2)

π‘†π‘ˆπ‘Ž arrivals in bad state. New π‘†π‘ˆπ‘Ž call/request is queued in 𝑄1

same as (6)

140 Table 6.3

External transitions from present state πœ™ = {πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2} of CAS+Q A π‘†π‘ˆπ‘Ž Departure

No. Next state Possible State Events Transition

Rate π‘†π‘ˆπ‘Ž Departure SU rate (1) ( πœ”, 𝑉𝑔(π‘Ž,𝑏)βˆ’ 1, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘Ž departure. An π‘†π‘ˆπ‘Ž in good

state, queued in 𝑄1 uses the idle channel-slots (πœƒπ‘”).

π‘‰π‘”πœƒπ‘”πœ‡π‘†

(2) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏)βˆ’ 1, 𝑉𝑏(π‘Ž,𝑏), 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘Ž departure. An π‘†π‘ˆπ‘Ž in moderate state, queued in 𝑄1 uses the idle channel-slots (πœƒπ‘š).

π‘‰π‘šπœƒπ‘šπœ‡π‘ 

(3) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏)βˆ’ 1, 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘Ž departure. An π‘†π‘ˆπ‘Ž in bad state, queued in 𝑄1 uses the idle channel-slots (πœƒπ‘).

π‘‰π‘πœƒπ‘πœ‡π‘ 

(4) ( πœ”, 𝑉𝑔(π‘Ž,𝑏)βˆ’ 1, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘Ž departure. An π‘†π‘ˆπ‘ in good state, queued in 𝑄2 uses the idle channel-slots (πœƒπ‘”).

Same as (1) (5) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏)βˆ’ 1, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘Ž departure. An π‘†π‘ˆπ‘ in

moderate state, queued in 𝑄2 uses the idle channel-slots (πœƒπ‘š).

Same as (2) (6) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏)βˆ’ 1, 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘Ž departure. An π‘†π‘ˆπ‘ in bad

state, queued in 𝑄2 uses the idle channel-slots (πœƒπ‘).

Same as (3) (7) ( πœ”, 𝑉𝑔(π‘Ž)βˆ’ 1, . . 𝑉𝑔(𝑏)+ , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. A transmitting

π‘†π‘ˆπ‘ in good state with minimum no. of channel-slots (πœƒπ‘”π‘šπ‘–π‘›) uses all the idle channel-slot

Same as (1)

(8) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)βˆ’ 1, . , π‘‰π‘š(𝑏)+ , 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. A transmitting π‘†π‘ˆπ‘ in moderate state with minimum no. of channel-slots (πœƒπ‘π‘šπ‘–π‘›) uses all the idle -slot

Same as (2)

(9) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)βˆ’ 1, . . 𝑉𝑏(𝑏)+ , 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. A transmitting π‘†π‘ˆπ‘ in bad state with minimum no. of channel-slots (πœƒπ‘π‘šπ‘–π‘›) uses all the idle channel-slot.

Same as (3)

(10) πœ”, 𝑉𝑔(π‘Ž)βˆ’ 1, . .0, . .0, . , 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏, 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. other π‘†π‘ˆπ‘ in good state, uses the idle channel- slots (πœƒπ‘”π‘šπ‘–π‘›) to attain maximum bound πœƒπ‘”π‘šπ‘Žπ‘₯

Same as (1)

(11) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)βˆ’, . .0. .0. , π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. other π‘†π‘ˆπ‘ in moderate state, uses the idle channel-slots (πœƒπ‘šπ‘šπ‘–π‘›) to attain maximum bound πœƒπ‘šπ‘šπ‘Žπ‘₯

Same as (2)

(12) (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)βˆ’ 1, , . .0, . . , π‘‰π‘π‘šπ‘Žπ‘₯, 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. other π‘†π‘ˆπ‘ in bad state, uses the idle channel-slots

Same as (3)

141 Table 6.4

External transitions from present state πœ™ = {πœ”, 𝑉𝑔(π‘Ž,𝑏). . , π‘‰π‘š(π‘Ž,𝑏), . . 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2} of CAS+Q A π‘†π‘ˆπ‘ Departure

( πœƒπ‘π‘šπ‘–π‘›) to attain maximum bound πœƒπ‘π‘šπ‘Žπ‘₯

(13) ( πœ”, 𝑉𝑔(π‘Ž,𝑏)βˆ’ 1, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. New SUs in good state cannot use the channel- slots.

same as 1

(14) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏)βˆ’ 1, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. New SUs in moderate state cannot use the channel-slots.

same as 2

(15) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏)βˆ’ 1, 𝑄1, 𝑄2) π‘†π‘ˆπ‘Ž departure. New SUs in bad state cannot use the channel- slots.

same as 3

πœ‡π‘ = πœ‡π‘”, πœ‡π‘š, πœ‡π‘ i.e., SU service rate is same irrespective of the channel condition.

No. Next state Possible State Events SU

Transition Rate π‘†π‘ˆπ‘ Departure

(1) ( πœ”, 𝑉𝑔(π‘Ž,𝑏)βˆ’ 1, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘ departure. An π‘†π‘ˆπ‘Ž in good state, queued in 𝑄1 uses the idle channel- slots (πœƒπ‘”).

π‘‰π‘”πœƒπ‘”π‘šπ‘–π‘›πœ‡π‘ 

(2) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏)βˆ’ 1, 𝑉𝑏(π‘Ž,𝑏), 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘ departure. An π‘†π‘ˆπ‘Ž in moderate state, queued in 𝑄1 uses the idle channel-slots (πœƒπ‘š).

π‘‰π‘”πœƒπ‘šπ‘šπ‘–π‘›πœ‡π‘ 

(3) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏)βˆ’ 1, 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘ departure. . An π‘†π‘ˆπ‘Ž in bad state, queued in 𝑄1 uses the idle channel-slots (πœƒπ‘).

π‘‰π‘”πœƒπ‘π‘šπ‘–π‘›πœ‡π‘ 

(4) ( πœ”, 𝑉𝑔(𝑏)βˆ’ 1, . . 𝑉𝑔(π‘Ž)+ , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. An π‘†π‘ˆπ‘Žin good state, queued in 𝑄1 uses the all the idle channel-slots.

. same as 1 (5) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(𝑏)βˆ’ 1, . . π‘‰π‘š(π‘Ž)+ , 𝑉𝑏(π‘Ž,𝑏), 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. An π‘†π‘ˆπ‘Ž in

moderate state, queued in 𝑄1 uses the all the idle channel-slots.

same as 2

(6) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(𝑏)βˆ’ 1, . . 𝑉𝑏(π‘Ž)+ , 𝑄1βˆ’ 1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. An π‘†π‘ˆπ‘Ž in bad state, queued in 𝑄1 uses the all the idle channel-slots.

same as 3

(7) ( πœ”, 𝑉𝑔(π‘Ž), 𝑉𝑔(𝑏)βˆ’ 1, . . 𝑉𝑔(𝑏)+ , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. A π‘†π‘ˆπ‘ with (πœƒπ‘”π‘šπ‘–π‘›) in good state, queued in 𝑄2 uses the all the idle channel- slots.

same as 1

(8) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), π‘‰π‘š(𝑏)βˆ’ 1, π‘‰π‘š(𝑏)+ , 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. A π‘†π‘ˆπ‘ with (πœƒπ‘šπ‘šπ‘–π‘›) in moderate state, queued in 𝑄2 uses the all the idle channel-slots.

same as 2

142

(9) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), 𝑉𝑏(𝑏)βˆ’ 1, . . 𝑉𝑏(𝑏)+ , 𝑄1, 𝑄2βˆ’ 1) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. A π‘†π‘ˆπ‘ with (πœƒπ‘π‘šπ‘–π‘›) in bad state, queued in 𝑄2 uses the all the idle channel- slots.

same as 3

(10) πœ”, 𝑉𝑔(π‘Ž), . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯βˆ’ 1, 𝑉𝑔(𝑏)+ , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. A transmitting π‘†π‘ˆπ‘ in good state with minimum no. of channel-slots (πœƒπ‘”π‘šπ‘–π‘›) uses all the idle channel- slots.

same as 1

(11) πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), . . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯βˆ’ 1, π‘‰π‘š(𝑏)+ , 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. A transmitting π‘†π‘ˆπ‘ in moderate state with minimum no. of channel-slots (πœƒπ‘šπ‘šπ‘–π‘›) uses all the idle channel- slots.

same as 2

(12) (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), . . 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯βˆ’ 1, 𝑉𝑏(𝑏)+ , 𝑄1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. A transmitting π‘†π‘ˆπ‘ in bad state with minimum no. of channel-slots (πœƒπ‘šπ‘šπ‘–π‘›) uses all the idle channel- slots.

same as 3

(13) (πœ”, 𝑉𝑔(π‘Ž), . .0, . .0, . , 𝑉𝑔(𝑏)+ . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. Every other π‘†π‘ˆπ‘ in good state, uses the idle channel-slots (πœƒπ‘”π‘šπ‘–π‘›) to attain maximum bound πœƒπ‘”π‘šπ‘Žπ‘₯

same as 1

(14) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), . .0. .0. , π‘‰π‘š(𝑏)+ . . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. Every other π‘†π‘ˆπ‘ in good state, uses the idle channel-slots (πœƒπ‘šπ‘šπ‘–π‘›) to attain maximum bound πœƒπ‘šπ‘šπ‘Žπ‘₯

same as 2

(15) (πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), . .0, . .0. , 𝑉𝑏(𝑏)+ . . 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯, 𝑄1, 𝑄2) π‘†π‘ˆπ‘with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. Every other π‘†π‘ˆπ‘ in good state, uses the idle channel-slots (πœƒπ‘π‘šπ‘–π‘›) to attain maximum bound πœƒπ‘π‘šπ‘Žπ‘₯

same as 3

(16) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯βˆ’ 1, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. New SUs in good state cannot use the channel-slots.

same as 1

(17) ( πœ”, 𝑉𝑔(π‘Ž,𝑏) , π‘‰π‘š(π‘Žπ‘), , , π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯βˆ’ 1, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ Departure. New SUs in moderate state cannot use the channel-slots.

same as 2

(18) ( πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏). , . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯βˆ’ 1, 𝑄1, 𝑄2) π‘†π‘ˆπ‘ with πœƒπ‘›π‘šπ‘Žπ‘₯ departure. New SUs in bad state cannot use the channel-slots.

same as 3

143 Table 6.5

External transitions from present state πœ™ = {πœ”, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2} of CAS+Q A PU Arrival

No. Next state Possible State Events PU Transition

Rate PU Arrivals

(1) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. An idle channel- slot is available irrespective of the channel state.

πœ†π‘

(2) (πœ” + 1, 𝑉𝑔(π‘Ž), 𝑉𝑔(𝑏). . , 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯βˆ’, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘ in good state with πœƒπ‘”π‘šπ‘Žπ‘₯ is interrupted and adjust-down channel-slots.

[πœƒπ‘”π‘šπ‘Žπ‘₯𝑉𝑔(𝑏) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘

(3) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), π‘‰π‘š(𝑏)… π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘ in moderate state with πœƒπ‘šπ‘šπ‘Žπ‘₯ is interrupted and adjust-down channel-slots.

[πœƒπ‘šπ‘šπ‘Žπ‘₯𝑉𝑔(𝑏) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘ (4) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏) , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), 𝑉𝑏(𝑏). . , 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘ in bad

state with πœƒπ‘π‘šπ‘Žπ‘₯ is interrupted and adjust-down channel-slots.

[πœƒπ‘π‘šπ‘Žπ‘₯𝑉𝑔(𝑏) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘ (5) (πœ” + 1, 𝑉𝑔(π‘Ž), 𝑉𝑔(𝑏)π‘šπ‘–π‘›βˆ’, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. The pre-empted

π‘†π‘ˆπ‘ with πœƒπ‘”π‘šπ‘–π‘› in good state, is queued (𝑄2) and no spectrum adjustment.

[πœƒπ‘π‘šπ‘–π‘›π‘‰π‘”(𝑏)

𝑧(πœ™) βˆ’ πœ”] πœ†π‘ (6) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), π‘‰π‘š(𝑏)π‘šπ‘–π‘›βˆ’, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. The pre-empted

π‘†π‘ˆπ‘ with πœƒπ‘šπ‘šπ‘–π‘› in moderate state, is queued (𝑄2) and no spectrum adjustment.

[πœƒπ‘π‘šπ‘–π‘›π‘‰π‘š(𝑏) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘

(7) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏) , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), 𝑉𝑏(𝑏)π‘šπ‘–π‘›βˆ’, 𝑄1, 𝑄2+ 1) PU arrival. The pre-empted π‘†π‘ˆπ‘ with πœƒπ‘π‘šπ‘–π‘› in bad state, is queued ( 𝑄2 ) and no spectrum adjustment.

[πœƒπ‘π‘šπ‘–π‘›π‘‰π‘š(𝑏) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘ (8) (πœ” + 1, 𝑉𝑔(π‘Ž), 𝑉𝑔(𝑏)π‘šπ‘–π‘›βˆ’ 1. . , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. Pre-empted π‘†π‘ˆπ‘

with πœƒπ‘π‘šπ‘–π‘› in good state is forced terminated and no spectrum adjustment.

Same as (5)

(9) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), π‘‰π‘š(𝑏)π‘šπ‘–π‘› βˆ’ 1. . , 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. Pre-empted π‘†π‘ˆπ‘ with πœƒπ‘π‘šπ‘–π‘› in moderate state is forced terminated and no spectrum adjustment.

Same as (6)

(10) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), 𝑉𝑏(𝑏)π‘šπ‘–π‘›βˆ’ 1, . . , 𝑄1, 𝑄2+ 1) PU arrival. Pre-empted π‘†π‘ˆπ‘ with πœƒπ‘π‘šπ‘–π‘› in bad state is forced terminated and no spectrum adjustment.

Same as (7)

(11) (πœ” + 1, 𝑉𝑔(π‘Ž), . .0, . .0, . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. An π‘†π‘ˆπ‘ in good state is queued (𝑄2). while another π‘†π‘ˆπ‘ in good state

Same as (5)

144

uses the idle channel-slots to attain upper bound πœƒπ‘”π‘šπ‘Žπ‘₯ (12) ( πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), . .0. .0. , . . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. An π‘†π‘ˆπ‘ in

moderate state is queued (𝑄2). while another π‘†π‘ˆπ‘ in moderate state uses the idle channel-slots to attain upper bound πœƒπ‘”π‘šπ‘Žπ‘₯

Same as (6)

(13) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), . .0, . .0. , . . 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯, 𝑄1, 𝑄2+ 1) PU arrival. An π‘†π‘ˆπ‘ in bad state is queued (𝑄2). while another π‘†π‘ˆπ‘ in bad state uses the idle channel-slots to attain upper bound πœƒπ‘”π‘šπ‘Žπ‘₯

Same as (7)

(14) (πœ” + 1, 𝑉𝑔(π‘Ž), . .0, . .0, . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. An π‘†π‘ˆπ‘ in good state is forced terminated, while another π‘†π‘ˆπ‘ in good state uses the idle channel- slots to attain upper bound πœƒπ‘”π‘šπ‘Žπ‘₯

Same as (5)

(15) ( πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), . .0. .0. , . . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2+ 1) PU arrival. An π‘†π‘ˆπ‘ in moderate state is forced terminated, while another π‘†π‘ˆπ‘ in moderate state uses the idle channel-slots to attain upper bound πœƒπ‘”π‘šπ‘Žπ‘₯

Same as (6)

(16) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), . .0, . .0. , . . 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯, 𝑄1, 𝑄2+ 1) PU arrival. An π‘†π‘ˆπ‘ in bad state is forced terminated, while another π‘†π‘ˆπ‘ in bad state uses the idle channel- slots to attain upper bound πœƒπ‘”π‘šπ‘Žπ‘₯

Same as (7)

(18)

(πœ” + 1, 𝑉𝑔(π‘Ž)βˆ’ , . . , 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯βˆ’. , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2)

PU arrival. An π‘†π‘ˆπ‘Ž in good state is pre-empted and π‘†π‘ˆπ‘ with πœƒπ‘”π‘šπ‘Žπ‘₯ donate channel-slots.

[ πœƒπ‘”π‘‰π‘”(π‘Ž) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘

(19) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏)π‘‰π‘š(π‘Ž)βˆ’ , . . , π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯βˆ’. , 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. π‘†π‘ˆπ‘Ž in moderate state is pre-empted and π‘†π‘ˆπ‘ with πœƒπ‘šπ‘šπ‘Žπ‘₯ donate channel-slots.

[πœƒπ‘šπ‘‰π‘š(π‘Ž) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘ (20) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), 𝑉𝑏(π‘Ž)βˆ’ , . . , 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯βˆ’, 𝑄1, 𝑄2) PU arrival. π‘†π‘ˆπ‘Ž in bad state

is pre-empted and π‘†π‘ˆπ‘ with πœƒπ‘π‘šπ‘Žπ‘₯ donate channel-slots.

[ πœƒπ‘π‘‰π‘(π‘Ž) 𝑧(πœ™) βˆ’ πœ”] πœ†π‘ (21) (πœ” + 1, 𝑉𝑔(π‘Ž)βˆ’ 1, 𝑉𝑔(𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. π‘†π‘ˆπ‘Ž in good

state is forced terminated no spectrum

adjustment/adaptation.

Same as (18)

(22) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)βˆ’ 1, π‘‰π‘š(𝑏)𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. π‘†π‘ˆπ‘Ž in moderate state is forced terminated no spectrum

adjustment/adaptation.

Same as (19)

145

(23) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)βˆ’ 1, 𝑉𝑏(𝑏), 𝑄1, 𝑄2) PU arrival. π‘†π‘ˆπ‘Ž in bad state is forced terminated no spectrum

adjustment/adaptation.

Same as (20)

(24) (πœ” + 1, 𝑉𝑔(π‘Ž)βˆ’ 1, 𝑉𝑔(𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) PU arrival. π‘†π‘ˆπ‘Ž in good state is forced terminated, π‘†π‘ˆπ‘ in good state, queued in 𝑄2 uses all the idle channel- slots

…….

Same as (18)

(25) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)βˆ’ 1, π‘‰π‘š(𝑏)𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2βˆ’ 1) PU arrival. π‘†π‘ˆπ‘Ž in moderate state is forced terminated, π‘†π‘ˆπ‘ in moderate state, queued in 𝑄2 uses all the idle channel- slots …….……

Same as (19)

(26) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)βˆ’ 1, 𝑉𝑏(𝑏), 𝑄1, 𝑄2βˆ’ 1) PU arrival. π‘†π‘ˆπ‘Ž in bad state is forced terminated, π‘†π‘ˆπ‘ in bad state, queued in 𝑄2 uses all the idle channel-slots

…

Same as (20)

(27) (πœ” + 1, 𝑉𝑔(π‘Ž)βˆ’ 1, . . 𝑉𝑔(𝑏)+ , π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘Ž in good state is forced terminated, π‘†π‘ˆπ‘ with πœƒπ‘”π‘šπ‘–π‘› in good state, uses all the idle channel-slots….

Same as (18)

(28) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž)βˆ’ 1, . . π‘‰π‘š(𝑏)+ , 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘Ž in moderate state is forced terminated, π‘†π‘ˆπ‘ with πœƒπ‘šπ‘šπ‘–π‘› in good state, uses all the idle channel-slots.

Same as (19)

(29) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž)βˆ’ 1, . . 𝑉𝑏(𝑏)+ , 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘Ž in bad state is forced terminated, π‘†π‘ˆπ‘ with πœƒπ‘π‘šπ‘–π‘› in good state, uses all the idle channel-slots.

…….

Same as (20)

(30) (πœ” + 1, 𝑉𝑔(π‘Ž), . .0, . .0, . . 𝑉𝑔(𝑏)π‘šπ‘Žπ‘₯, π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘Ž in good state is pre-empted and π‘†π‘ˆπ‘ in good state is forced terminated. All other π‘†π‘ˆπ‘s in good state uses the idle channel-slots to attain πœƒπ‘”π‘šπ‘Žπ‘₯

…

Same as (18)

(31) ( πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž), . .0. .0. , . . π‘‰π‘š(𝑏)π‘šπ‘Žπ‘₯, 𝑉𝑏(π‘Ž,𝑏), 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘Ž in moderate state is pre-empted and π‘†π‘ˆπ‘ in moderate state is forced terminated. All other π‘†π‘ˆπ‘s in good state uses the idle channel-slots to attain πœƒπ‘šπ‘šπ‘Žπ‘₯…

Same as (19)

146

(32) (πœ” + 1, 𝑉𝑔(π‘Ž,𝑏), π‘‰π‘š(π‘Ž,𝑏), 𝑉𝑏(π‘Ž), . .0, . .0. , . . 𝑉𝑏(𝑏)π‘šπ‘Žπ‘₯, 𝑄1, 𝑄2) PU arrival. An π‘†π‘ˆπ‘Ž in bad state is pre-empted and π‘†π‘ˆπ‘ in bad state is forced terminated. All other π‘†π‘ˆπ‘s in good state uses the idle

channel-slots to

attain πœƒπ‘π‘šπ‘Žπ‘₯…

Same as (20)

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