1 Introduction
2 Related works
3 General model of a switching network
3.1 Structure of offered traffic
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\(y_{\text {En},j,c}(n)\)—the average number of calls of class c that are generated by Engset sources from set \({\mathbb {Z}}_{\text {En},j}\) and serviced in the system in the occupancy state of n AUs,
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\(y_{\text {Pa},k,c}(n)\)—the average number of calls of class c that are generated by Pascal sources from set \({\mathbb {Z}}_{\text {Pa},k}\) and serviced in the system in the occupancy state of n AUs,
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\(\alpha _{\text {En},j}\)—the average intensity of traffic offered by a single Engset source from set \({\mathbb {Z}}_{\text {En},j}\):where \(\gamma _{\text {En},j}\)—intensity of generating calls by a single free Engset source that belongs to set \({\mathbb {Z}}_{\text {En},j}\),$$\begin{aligned} \alpha _{\text {En},j}=\sum \limits _{c=1}^{c_{\text {En},j}}\eta _{\text {En},j,c}{\gamma _{\text {En},j}}/{\mu _{c}}, \end{aligned}$$(4)
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\(\beta _{\text {Pa},k}\)—the average intensity of traffic offered by a single Pascal source that belongs to set \({\mathbb {Z}}_{\text {Pa},k}\):where \(\gamma _{\text {Pa},k}\)—intensity of call generation by a single free Pascal source that belongs to set \({\mathbb {Z}}_{\text {Pa},k}\).$$\begin{aligned} \beta _{\text {Pa},k}=\sum \limits _{c=1}^{c_{\text {Pa},k}}\eta _{\text {Pa},k,c}{\gamma _{\text {Pa},k}}/{\mu _{c}}, \end{aligned}$$(5)
3.2 Algorithms for setting up connections in a switching network
4 Models of output links of a switching network
4.1 Introduction
4.2 Model of output links with resource reservation
4.3 Model of an output group with threshold mechanisms
5 Models of inter-stage links in a switching network
5.1 Introduction
5.2 A new model of inter-stage links with threshold mechanisms
6 Recurrent method for the determination of blocking probability in a switching network
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\(d_z(c)\)—effective availability for traffic stream of class c in equivalent z-stage switching network,
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\(\pi _z(c)\)—the so-called probability of direct unavailability of a given switch of the last stage for a call of class c. Parameter \(\pi _z(c)\) determines the probability of an event in which a connection of class c between a given switch of the first stage and a given switch of the last stage cannot be set up. The determination of this parameter is based on the analysis of a channel graph for the equivalent switching network [32, 59],
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\(\upsilon \)—the number of output links in a given direction,
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\(\eta \)—the value that defines which part of fictitious traffic, in a switch of the first stage, is serviced by the considered direction. If traffic is uniformly distributed between all \(\upsilon \) directions, we get \(\eta =1/\upsilon \).
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\(E_1(a(c))\) is the blocking probability in inter-stage links of the switching network (between stages 1 and 2), determined for a group with the capacity of 1 BBU to which traffic a(c) is offered,
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e(c) is a fictitious load for calls of class c in inter-stage links (between stages 1 and 2). This load is equal to the blocking probability for calls of class c in an inter-stage link of the real network, modelled by a full-availability group with multiservice traffic sources (Sect. 5).
PGPPBRec-U method | |
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2. Initialization of iteration step \(l=1\)
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3. Calculation of fictitious load in inter-stage links of switching networks on the basis of a generalised model of inter-stage links—Formula (44) | |
4. Calculation of the value of offered traffic—Formula (52) | |
5. Calculation of blocking probability in a single switch—Formula (48) | |
6. Increase of iteration step: \(l=l+1\)
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7. Determination of effective availability parameter \(d_l(c)\) in l-stage switching network with multiservice traffic sources—Formula (46) | |
8. Determination of internal blocking probability in the switching network for individual call classes–Formula (54) | |
9. Repetition of steps 7–9: | |
point-to-group internal blocking, iterative process terminates when \(l=z\), | |
point-to-point internal blocking, iterative process terminates when \(l=z-1\)
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10. Calculation of total blocking probability—Formula (45) |
7 Numerical results
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System 1—Threshold mechanism:
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structure of switching network: \(\upsilon =4\), \(f=42\) AUs, \(V=168\) AUs;
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structure of offered traffic:
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traffic classes: \(m=4\), \(t_{1,0}=1\) AU, \(\mu _{1,0}^{-1}=1\), \(t_{2,0}=6\) AUs, \(\mu _{2,0}^{-1}=1\), \(t_{3,0}=10\) AUs, \(\mu _{3,0}^{-1}=1\), \(t_{3,1}=8\) AUs, \(\mu _{3,0}^{-1}=1,25\), \(t_{4,0}=12\) AUs, \(\mu _{4,0}^{-1}=1\), \(t_{4,1}=8\) AUs, \(\mu _{4,1}^{-1}=1,5\);
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sets of traffic sources: \(S=2\), \({\mathbb {C}}_{\text {Pa},1}=\{1,2,3\}\), \(\eta _{\text {Pa},1,1}=0,6\), \(\eta _{\text {Pa},1,2}=0,2\), \(\eta _{\text {Pa},1,3}=0,2\), \(S_{\text {Pa},1}=950\), \({\mathbb {C}}_{\text {En},2}=\{1,2,4\}\), \(\eta _{\text {En},2,1}=0,6\), \(\eta _{\text {En},2,2}=0,2\), \(\eta _{\text {En},2,4}=0,2\), \(N_{\text {En},2}=950\);
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threshold mechanism: \(Q_{3,1}=Q_{4,1}=126\) AUs.
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System 2—Threshold mechanism:
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structure of switching network: \(\upsilon =4\), \(f=28\) AUs, \(V=112\) AUs;
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structure of offered traffic:
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traffic classes: \(m=4\), \(t_{1,0}=1\) AU, \(\mu _{1,0}^{-1}=1\), \(t_{2,0}=2\) AUs, \(\mu _{2,0}^{-1}=1\),\(t_{3,0}=4\) AUs, \(\mu _{3,0}^{-1}=1\), \(t_{4,0}=8\) AUs, \(\mu _{4,0}^{-1}=1\), \(t_{4,1}=4\) AUs, \(\mu _{4,1}^{-1}=2\), \(t_{5,0}=12\) AUs, \(\mu _{5,0}^{-1}=1\), \(t_{5,1}=8\) AUs, \(\mu _{5,1}^{-1}=1,5\), \(t_{5,2}=4\) AUs, \(\mu _{5,2}^{-1}=3\);
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sets of traffic sources: \(S=3\), \({\mathbb {C}}_{\text {Er},1}=\{1,3\}\), \(\eta _{\text {Er},1,1}=0,6\), \(\eta _{\text {Er},1,3}=0,4\), \({\mathbb {C}}_{\text {En},2}=\{1,2,4\}\), \(\eta _{\text {En},2,1}=0,5\), \(\eta _{\text {En},2,2}=0,3\), \(\eta _{\text {En},2,4}=0,2\), \(N_{\text {En},2}=1000\), \({\mathbb {C}}_{\text {En},3}=\{1,4,5\}\), \(\eta _{\text {En},3,1}=0,6\), \(\eta _{\text {En},3,4}=0,1\), \(\eta _{\text {En},3,5}=0,3\), \(N_{\text {En},3}=1000\);
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threshold mechanism: \(Q_{4,1}=Q_{5,1}=84\) AUs, \(Q_{5,2}=72\) AUs.
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System 3—Bandwidth reservation mechanism:
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structure of switching network: \(\upsilon =4\), \(f=36\) AUs, \(V=144\) AUs;
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structure of offered traffic:
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traffic classes: \(m=3\), \(t_{1}=1\) AU, \(\mu _{1}^{-1}=1\), \(t_{2}=4\) AUs, \(\mu _{2}^{-1}=1\),\(t_{3}=10\) AUs, \(\mu _{3}^{-1}=1\);
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sets of traffic sources: \(S=2\), \({\mathbb {C}}_{\text {Er},1}=\{1,2,3\}\), \(\eta _{\text {Er},1,1}=0,7\), \(\eta _{\text {Er},1,2}=0,2\), \(\eta _{\text {Er},1,3}=0,1\), \({\mathbb {C}}_{\text {En},2}=\{1,2,3\}\), \(\eta _{\text {En},2,1}=0,6\), \(\eta _{\text {En},2,2}=0,2\), \(\eta _{\text {En},2,3}=0,2\), \(N_{\text {En},2}=900\),
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threshold mechanism: \(R_{1}=R_{2}=119\) AUs.
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System 4—Bandwidth reservation mechanism:
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structure of switching network: \(\upsilon =4\), \(f=34\) AUs, \(V=136\) AUs;
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structure of offered traffic:
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traffic classes: \(m=4\), \(t_{1}=1\) AUs, \(\mu _{1}^{-1}=1\), \(t_{2}=4\) AUs, \(\mu _{2}^{-1}=1\), \(t_{3}=6\) AUs, \(\mu _{3}^{-1}=1\), \(t_{4}=10\) AUs, \(\mu _{4}^{-1}=1\);
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sets of traffic sources: \(S=2\), \({\mathbb {C}}_{\text {Pa},1}=\{1,2,4\}\), \(\eta _{\text {Pa},1,1}=0,7\), \(\eta _{\text {Pa},1,2}=0,2\), \(\eta _{\text {Pa},1,4}=0,1\), \(S_{\text {Pa},1}=900\), \({\mathbb {C}}_{\text {En},2}=\{1,3,4\}\), \(\eta _{\text {En},2,1}=0,6\), \(\eta _{\text {En},2,3}=0,2\), \(\eta _{\text {En},2,4}=0,2\), \(N_{\text {En},2}=900\);
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threshold mechanism: \(R_1=R_2=R_3=112\) AUs.
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